DEFORMATION THEORY - boun.edu.tr · 1.1.2. The category of schemes, or the category of schemes over...

DEFORMATION THEORY

W. D. GILLAM

Abstract. These are lecture notes for a short class on deformation theory (28hours of lectures) given at ETH Zürich in the fall of 2012.

Contents

1. Introduction 5

1.1. Spaces 5

1.2. Moduli problems 5

1.3. Thickenings 6

1.4. Questions 6

1.5. Linearization 7

1.6. Deformation/obstruction theory 9

1.7. Obstruction theory for invertible sheaves 10

1.8. Algebraization 11

2. Innitesimal Thickenings 11

2.1. Topological invariance 12

2.2. Square-zero thickenings 13

2.3. Flatness 15

2.4. Descent of properties of morphisms 18

2.5. Descent of properties of modules 22

3. Dierentials and Algebra Extensions 24

3.1. Denitions of dierentials 24

3.2. Sheacation 26

3.3. Algebra extensions 27

3.4. Adjointness property 29

3.5. Derivations 30

Date: February 7, 2013.1

DEFORMATION THEORY 2

3.6. Direct limit preservation 31

3.7. Exact sequences 32

3.8. Geometric dierentials 35

3.9. The case of schemes 36

4. Commutative algebra 37

4.1. Artinian rings 37

4.2. Regular sequences 38

4.3. Regular rings 39

4.4. Flat, projective, and free modules 40

4.5. Fiberwise criteria 42

5. Formal smoothness 45

5.1. Denitions and rst properties 46

5.2. Examples 51

5.3. Formal smoothness and dierentials 52

5.4. The formal Jacobian criterion 55


5.6. The smooth locus is open 59

5.7. Formal smoothness and eld extensions 60

5.8. The Jacobian criterion 63

5.9. Artinian lifting property 66

5.10. Characterization of smoothness 71

6. The Truncated Cotangent Complex 73

6.1. Flat modules 74

6.2. Symmetric algebra 75

6.3. Functoriality of LB/A 78

6.4. Comparison theorem 79

6.5. Flat base change 84

7. The Truncated Cotangent Complex Continued 85

7.1. Transitivity triangle 85

7.2. The Fundamental Theorem 88

7.3. Curvilinear algebra extensions 94



7.5. Smoothness and the cotangent complex 96

7.6. Representable obstruction theory 99

7.7. Obstruction theory revisited 101

8. The Cotangent Complex 102

8.1. Denition of the cotangent complex 102

8.2. Basic properties 105

8.3. Regular embeddings and l.c.i. morphisms 106

9. Deformation of schemes 107

9.1. Obstruction theory 107

9.2. Automorphisms of curves 109

9.3. Moduli of nodal curves 110

10. Deformation of quotients 113

10.1. Abstract deformation problem 113

10.2. Quot schemes 116

10.3. Obstruction theory 117

10.4. Smoothness of the Grassmannian 120

10.5. Smoothness of some Quot schemes 121

11. Hilbert schemes of points 124

11.1. Conguration space 124

11.2. Fogarty's smoothness theorem 125

11.3. The punctual Hilbert scheme 128

12. Pathologies 131

12.1. Rigid schemes 131

12.2. Formally smoothable schemes 132

12.3. Nonsmoothable schemes 134

12.4. Iarrobino's Example: Hilb96 A3 134

13. Homological Algebra 134

13.1. Yoneda Ext 135

13.2. Yoneda pairing 137

13.3. Yoneda Ext is universal 138

13.4. Ext comparison 141

13.5. Sheaf Ext 144


13.6. Quasi-coherent sheaf Ext 150

13.7. Categories of complexes 155

13.8. Hyper Ext 159

14. Groupoid Fibrations 164

14.1. Examples 165

14.2. 2-categories 166

14.3. Groupoid brations as a 2-category 167

14.4. 2-commutative diagrams 171

14.5. 2-bered products 174

Exercises 175

References 185


1. Introduction

A major goal in algebraic geometryand in other eldsis to understand variousmoduli spaces or moduli problems. A rigorous denition of moduli problem" iscategory bered in groupoids"M→ Esp. We will explain this denition carefullylater but for now it is good enough to have in mind the study of some algebraic"objects where there is a reasonable notion of pullback".

1.1. Spaces. The base category Esp of the groupoid bration will generally besome category of spaces" such as:

1.1.1. The category of locally ringed spaces, or the category of locally ringed spacesover some xed base.

1.1.2. The category of schemes, or the category of schemes over some xed base.

1.1.3. The catgeory of analytic spaces, or analytic spaces over a xed base.

There are more exotic" examples like dierentiable spaces," log schemes," etc.Later on it will be convenient to have some kind of topology" on the category Esp,but we can ignore this for now. It will also be important in 1.5 to have somemaps of spaces called square-zero closed embeddings" (as we do in every examplementioned so far). For the sake of concreteness, however, we will now assume thatEsp is the category Sch of schemes.

1.2. Moduli problems. Here are some typical examples of moduli problemsM:

1.2.1. The category Sch/X of maps to some xed scheme X. Moduli problems ofthis type are called representable.

1.2.2. Moduli of at families of schemes.

1.2.3. A variant of the previous moduli problem is the study of at families ofschemes having other additional properties (e.g. at and proper families, at andproper families whose geometric bers are .... (e.g. nodal curves)).

1.2.4. Invertible sheaves (line bundles). A variant is to consider moduli of linebundles on a xed scheme X.

1.2.5. More generally, one can study moduli of torsors (principal bundles) for otheralgebraic groups (a line bundle is a Gm-torsor). Of course there is a variant whereone studies moduli of torsors on a xed scheme X.

1.2.6. In particular, one can study moduli of vector bundles, or moduli of vectorbundles on some xed scheme X.


1.2.7. Coherent sheaves. Like principal bundles, this moduli problem also containsvector bundles as a specialization.

1.2.8. Moduli of maps between two given schemes X and Y .

1.2.9. Moduli of (at families of) closed subschemes of a given scheme X.

It might happen that one moduli problem falls into several dierent categoriesabove, in which case we will potentially have several dierent ways of studying thatproblem. For example, it might be that the moduli space of closed subschemes ofsome given scheme X is representable.

In studying such moduli problems there are (at least) two major tools: DescentTheory and Deformation Theory. The goal of the rst is to understand to whatextent one can glue together locally dened objects (and morphisms). That is,descent theory seeks to determine which groupoid brations are stacks in whichtopologies.

Deformation theorythe subject of these notesattempts (in large part) to ad-dress the following question: Suppose one has a closed embedding Y → Y ′ ofschemes (or schemes over some base, or some other kind of spaces...) and an objectm of our moduli problem over Y . What can we say about the objects m′ of ourmoduli problem over Y ′ that pull back (restrict) to m?

1.3. Thickenings. Often not much can be said at this level of generality, so wemight look at some particularly nice Y ′ → Y such as the following:

1.3.1. It is very common to look at the case where Y = Spec k for a eld k andY ′ = SpecA for a local Artinian ring A with residue eld k. A variant is to letY → Y ′ be Spec of a surjection of local Artinian rings (often with the same residueeld) A′ → A.

1.3.2. If we work with schemes over a eld k, then Y → Y ′ could be a k-point of asmooth (and connected) curve Y ′ over k.

1.3.3. The usual algebraic variant of the previous situation is where Y ′ = SpecA,where (A,m) is a discrete valuation ring, and Y is (the spectrum of) the residueeld k = A/m of A. The Grothendieck school likes to use trait" (from the Frenchfor line") for the spectrum of a d.v.r. (c.f. [SGA7 I.0.0.1]). One can further dividethese kind of thickenings into dierent cases, according to whether A is m-adicallycomplete, or of mixed characteristic, etc. Keep in mind the following possibilitiesfor A: k[x](x), k[[x]], Z(p), and the p-adic integers Zp.

1.4. Questions. The sort of questions we might ask will depend on our moduliproblemM. Here are some typical examples:


1.4.1. In (1.2.1), m is a morphism of schemes m : Y → X. We might ask whetherthis extends to a morphism m′ : Y ′ → X. If so, then we might try to understandthe set of all such extensions. Maybe the extension is unique, for example.

1.4.2. In (1.2.2) with Y → Y ′ as in (1.3.2), m is, say, a variety over k and we mightask whether there is an m′ (a scheme at over Y ′) such that the ber at some otherk-point of m′ is a smooth k-variety. That is, we ask: Is m smoothable?" Of coursethis has an analog with Y → Y ′ as in (1.3.3) where we ask whether there is an m′

over the trait Y ′ with special ber m and generic ber a smooth variety over thefraction eld of the d.v.r.

1.4.3. In (1.2.2) with Y → Y ′ as in (1.3.3) with the d.v.r. of mixed characteristic,we ask whether a given scheme over the positive characteristic residue eld lifts tocharacteristic zero.

1.4.4. In (1.2.6) with Y → Y ′ as in (1.3.2), m is a vector bundle on a variety Xand we might ask whether there is a vector bundle E ′ on X × Y ′ (i.e. an m′) whoserestriction to X × y′ is, say, split. Of course one can ask very general questionsabout reduction of structure group for principal bundles along these lines.

1.4.5. In (1.2.9) there is a variant of the smoothability question from (1.4.2) wherewe ask whether a given closed subscheme Z ⊆ X is ambiently smoothable". Or wemight simply ask whether one closed subscheme lies in a (at) family with another.

It is often dicult to answer questions like this all at once". The approach ofdeformation theory is to divide these lifting questions into two parts:

1.5. Linearization. The rst step is to consider a linearized" version of the liftingproblem. Usually in mathematics, linearizing" a problem means ignoring terms oforder greater than one (squares and cubes and so forth) and what we do here is nodierent: We look only at the rst innitesimal neighborhood of Y in Y ′ (the closedsubscheme of Y ′ dened by the square of the ideal I dening Y ) and we consideronly the problem of lifting to this neighborhood. Equivalently, we study our liftingproblem only when Y → Y ′ is dened by a square-zero ideal.

Such square zero thickenings" come in two avors: if the inclusion Y → Y ′ has aretract r : Y ′ → Y , then the thickening is called trivial (otherwise it is nontrivial).If Y → Y ′ is a trivial thickening with a xed retract r : Y ′ → Y , then any object mof our moduli problem over Y has a trivial lifting to an object m′0 over Y

′: we justpull back m along the retract r. However, if the thickening Y → Y ′ is nontrivial,there may or may not be any m′ lifting m.

Square zero thickenings form a category ExSch where morphisms are commuta-tive squares. There is a related category SchMod whose objects are pairs (Y, I)consisting of a scheme Y and a (quasi-coherent) OY -module I where a morphism


(Y, I)→ (Z, J) is a pair (f, g) consisting of a map of schemes f : Y → Z and a mapof OY -modules g : f ∗J → I. There is a forgetful functor

ExSch → SchMod(1.5.1)(Y → Y ′) 7→ (Y, I)

taking a square-zero thickening Y → Y ′ to the pair (Y, I) consisting of the scheme Yand the ideal I of Y in Y ′ (with its natural structure of quasi-coherent OY -module).

Once we understand liftings of m over the rst innitesimal neighborhood Y1 ofY in Y ′, we will study how each such lifting can (or can't) be lifted to the secondinnitesimal neighborhood Y2 of Y in Y ′ (note Y1 → Y2 is again a square-zeroclosed embedding), then how each such lift can be lifted to the third innitesimalneighborhood Y3 and so forth. The idea is to work our way up to larger and largerinnitesimal neighborhoods one linear" step at a time.

1.5.1. For example, if Y → Y ′ was as in (1.3.2) we will instead consider only (Specof) the closed embedding k[x]/x2 → k. We will then work our way up the square-zero thickenings k[x]/xn+1 → k[x]/xn. Notice that the rst (n = 1) thickening istrivial, but the further thickenings are not.

1.5.2. In the mixed characteristic variant of (1.3.3) we might look Y → Y ′ given by(Spec of) Z/p2Z→ Z/pZ, then work our way up the thickenings Z/pn+1Z→ Z/pnZ.Notice that none of these thickenings is trivial.

The advantage of this linearized problem is that one can reasonably hope to under-stand the lifting problem entirely," often in terms of cohomology groups associatedto Y → Y ′ and m. This is the subject of deformation/obstruction theory. Beforesaying something about this in 1.6, we again mention some of the specic liftingquestions one might consider:

1.5.3. In (1.2.1), we ask whether a given map of schemes m : Y → X can beextended to a map m′ : Y ′ → X. If we quantify over all square-zero thickeningY → Y ′ and all m : Y → X we are asking whether X is (formally) smooth.

In general, an understanding of our linearized lifting problem is enough to un-derstand whether a moduli problem is formally smooth". This is just a matter ofdenitions, though the point is that for a moduli problem represented by a (locallynitely presented) scheme, the abstract notion of formally smooth" coincides withthe usual notion of smooth" in algebraic geometry, as we will see later.

1.5.4. In (1.2.2) we might ask about liftings of a given variety X over a eld k toa scheme at over k[x]/x2. Notice that there is at least one such lifting, namelyX ×Spec k Spec k[x]/x

2, because our thickening here is trivial. But there might beother liftings of X not of this form. Such liftings of X are often called rst orderdeformations of X.


1.5.5. In the mixed characteristic variant of (1.3.3) we might ask whether a givenvariety X over Fp = Z/pZ lifts to a scheme at over Z/p2Z. That is: Does Xlift mod p2? Already this is a surprisingly rich question related to the existence ofFrobenius splittings and degeneration of the Hodge-to-de Rham spectral sequence.

Notice that it is not at all clear how we can tackle problems like (1.4.2), (1.4.4),and (1.4.5) using only the linearized approach. But we might very quickly be ableto give (negative) answers to problems like (1.4.1) and (1.4.3).

1.6. Deformation/obstruction theory. In favorable situations, the linearizeddeformation theory of a moduli problem M problem culminates in an deforma-tion/obstruction theory (or Aut/Def/Ob theory) forM, by which we mean: to eachtriple (Y, I,m) consisting of an object (Y, I) of SchMod and an object m of Mover Y , there are three (functorially) associated OY (Y )-modules

Aut(Y, I,m), Def(Y, I,m), Ob(Y, I,m)

(called the automorphism group, the deformation group, and the obstruction group)and to each pair (Y → Y ′,m) consisting of a square-zero thickening Y → Y ′ withideal I and an object m ofM over Y , there is an element

ob(Y → Y ′,m) ∈ Ob(Y, I,m)

(called the obstruction class) such that:

1.6.1. There exists an object m′ ofM over Y ′ lifting m i ob(Y → Y ′,m) = 0.

1.6.2. If ob(Y → Y ′,m) = 0, there is a torsorial action of Def(Y, I,m) on the set ofisomorphism classes of lifts m′ of m to Y ′.

1.6.3. If ob(Y → Y ′,m) = 0, then the automorphism group of any lift m′ of m (overm and Y ′) is isomorphic to Aut(Y, I,m).

Just to be clear here: The Aut/Def/Ob modules themselves depend only on (Y, I),while the obstruction class depends on an actual choice of square-zero thickeningY → Y ′ with ideal I. Everything is supposed to be functorial in (Y → Y ′,m)including the obstruction class, the torsorial action in (1.6.2), and (when it vanishes),and the isomorphism in (1.6.3).

Notice that (as an abelian group at least), the automorphism group Aut(Y, I,m)is determined (up to unique isomorphism) by the moduli problemM itself. (Becauseone can always take Y → Y ′ to be the trivial square-zero extension of Y by I sothat ob(Y → Y ′,m) = 0.) The deformation group Def(Y, I,m) is less tightly relatedto M, but at least the cardinality of Def(Y, I,m) is determined by M in light of(1.6.2). There is, however, considerable freedom" to vary the obstruction group.This is related to the fact that a given moduli problem might be viewed in manydierent ways. The OY (Y )-module structure on the Aut/Def/Ob groups seems tocome out of nowhere, but it exists in practice.


It may seem highly unlikely that any reasonable moduli problem would admit anAut/Def/Ob theory. Even if our moduli problem is representable (1.2.1), it is notat all obvious that it should admit an Aut/Def/Ob theory, though in fact this is thecase. Even if the representing scheme X is smooth it takes a certain amount of workto make an Aut/Def/Ob theory. In later lectures, we will construct Aut/Def/Obtheories for all (or almost all) of the moduli problems mentioned in the beginningof this lecture.

Remark 1. If you drop the atness requirement in the moduli problem (1.2.2), theresulting moduli problem won't have an Aut/Def/Ob theory in the above sense.

1.7. Obstruction theory for invertible sheaves. For now, we will content our-selves with the construction of an Aut/Def/Ob theory for the moduli problem ofinvertible sheaves (1.2.4). To be clear about this moduli problem, we are lookingat the category M whose objects are pairs (Y,m) consisting of a scheme Y andan invertible sheaf m on Y and whose morphisms (Y,m) → (Z, n) are pairs (f, g)consisting of a map of schemes f : Y → Z and an isomorphism g : f ∗n → m ofOY -modules; there is a forgetful functorM→ Sch given by (Y,m) 7→ Y .

The construction of an Aut/Def/Ob theory for this M is even easier than theconstruction of an Aut/Def/Ob theory for (general) representable moduli problems!We can take

Aut(Y, I,m) := H0(Y, I)

Def(Y, I,m) := H1(Y, I)

Ob(Y, I,m) := H2(Y, I).

(Notice that in this case there is not even a dependence on the line bundle m onY , though this is a rather special feature of this moduli problem.) We still have toconstruct the obstruction class ob(Y → Y ′,m) for a square-zero thickening Y → Y ′

with ideal I and a line bundle m on Y . We have a short exact sequence of sheavesof abelian groups

0→ I → O∗Y ′ → O∗Y → 0

on the common topological space of Y and Y ′ (c.f. Lemma 2.1.1) where the leftmap is i 7→ 1 + i. The point is that we can think of a line bundle m on Y (up toisomorphism) as an element of H1(Y,O∗Y ) and a line bundle m′ on Y ′ lifting m (andup to isomorphism) as an element m′ ∈ H1(Y,O∗Y ′) mapping to m under the map

H1(Y ′,O∗Y ′) → H1(Y,O∗Y ).

We let ob(Y → Y ′,m) be the image of m under the connecting homomorphism

H1(Y,O∗Y ) → H2(Y, I)

in the long exact sequence of cohomology groups associated to the above shortexact sequence of sheaves. All the desired properties of the Aut/Def/Ob theory areobtained by looking at the aforementioned long exact sequence.


1.8. Algebraization. Suppose (A,m, k) is a (noetherian) local ring. Let Yi :=Spec(A/mi+1), so that we have a sequence of square-zero closed embeddings

Spec k = Y0 → Y1 → Y2 → · · · .Suppose we also have a sequence of objects of our moduli problemM

m0 → m1 → · · ·lifting this sequence of square-zero closed embeddings. (Note that the descriptionof all possible such m• was achieved" in our linearized study.) The rst questionis whether m• is eective: Is there an object m ofM over (Spec of) the completelocal ring A so that m• is pulled back from m? Questions of eectivity can oftenbe answered armatively using Grothendieck's Existence Theorems ; we will discussthis later.

Even if m• is eective, we still don't know whether it is actually obtained froman object of our moduli problem over the original local ring (A,m, k). In fact thisisn't really quite the right question to ask; for one thing this question is trivial if Ais already m-adically complete. A better" question is whether there is some otherlocal ring (of essentially nite type") (B,m) with the same completion as A (andhence the same quotients B/mn+1 = A/mn+1) and an object m of M over SpecBrestricting to m•. This is roughly the question addressed by Artin's AlgebraizationTheorems. We will of course make this discussion much more precise later in thelectures.

2. Infinitesimal Thickenings

As we began to discuss in 1.5, square-zero thickenings play an important rolein deformation theory. In this section we will collect some basic facts about suchthickenings for later use. Many of the results hold for more general kinds of nilpotentthickenings."

Denition 1. An ideal I in a ring (or sheaf of rings) A (on a space X) is called......sqaure-zero i I2 = 0 (this can be checked locally or on stalks). ....nilpotent iIn = (0) for some n ≥ 0. ...locally nilpotent i X has a cover Ui so that I|Ui

⊆ A|Ui

is nilpotent for each i. ...stalkwise nilpotent i Ix ⊆ Ax is nilpotent for each x ∈ X.

Denition 2. A closed embedding of schemes (or locally ringed spaces) X → Ywith ideal I ⊆ OY is called square-zero (resp. locally nilpotent, stalkwise nilpotent)i I is square-zero (resp. locally nilpotent, stalkwise nilpotent). We sometimessay that a map of schemes f : X → Y is a square-zero (resp. locally nilpotent,stalkwise nilpotent) thickening i f is a closed embedding with square-zero (resp.locally nilpotent, stalkwise nilpotent) ideal sheaf.

Note that each of these nilpotence conditions" on I implies the one listed next.The last two conditions are equivalent under some mild assumptions (Exercise 1)in particular for locally nitely generated ideal sheaves on schemes (hence for allquasi-coherent ideal sheaves on a locally noetherian scheme).


Lemma 2.0.1. Let f : A→ B, g : B → C be ring homomorphisms. If f and g aresurjective with nilpotent kernels, so is gf : A → C. If gf : A → C and f : A → Bare surjective with nilpotent kernels, so is g. If f : A→ B is a surjective ring mapwith nilpotent (resp. square-zero) kernel, then any pushout of f is also a surjectivering map with nilpotent (resp. square-zero) kernel.

Proof. This is an easy exercise with the denitions.

Remark 2. If f : X → Y is a closed embedding of schemes, then any base changeof f (calculated in the category of schemes) is the same as the corresponding basechange calculated in ringed spaces [G, Corollary 11]in particular, the underlyingtopological space of the base change is the base change of the underlying spaces,so a closed embedding of schemes which is also a homeomorphism is a universalhomeomorphism. The same remark holds with scheme" replaced everywhere bylocally ringed space".

Lemma 2.0.2. Let f : X → Y and g : Y → Z be maps of schemes. If f and gare stalkwise-nilpotent thickenings (resp. locally nilpotent thickenings), so is gf . Ifg and gf are stalkwise-nilpotent thickenings (resp. locally nilpotent thickenings), sois f . Stalkwise-nilpotent thickenings, locally nilpotent thickenings, and square-zerothickenings are stable under base change.

Proof. The rst statements are clear from the rst part of Lemma 2.0.1. The stabilityunder base change statement is clear from the stability under pushout statements inLemma 2.0.1 (for stability of stalkwise-nilpotent thickenings under pushout it helpsto use the fact that the base change is the one calculated in ringed spaces, as inRemark 2).

2.1. Topological invariance.

Lemma 2.1.1. Let f : X → Y be a stalkwise nilpotent thickening of schemes. Thenf is a universal homeomorphism on topological spaces.

Proof. Since stalkwise nilpotent thickenings are stable under base change (Lemma 2.0.2)it is enough to prove f is a homeomorphism. Since f is a closed embedding on topo-logical spaces it suces to prove that f is surjective. As with any closed embeddingf , we have a short exact sequence of (quasi-coherent) OY -modules

0→ I → OY → f∗OX → 0

where I ⊆ OY is the ideal of X in Y . The support of f∗OX is the image of theclosed embedding f . But for any y ∈ Y , we then have an exact sequence

0→ Iy → OY,y → (f∗OX)y → 0,

and Iy ⊆ OY,y is nilpotent, so it must be contained in the maximal ideal my ⊆ OY,y,hence the quotient (f∗OX)y is not zero, hence y is in the image of f .


Remark 3. In Lemma 2.1.1, one can also conclude that f induces an equivalencebetween the étale topos of X and that of Y . See [SGA4, VIII.1.1] or [Milne, The-orem 3.23]. (Note that a closed embedding is certainly an integral and radicielmorphism. There will be some unnecessary noetherian hypotheses in the Milnebook, but one can get a more complete proof from scratch in less time.) In fact itis shown in [EGA, IV.18.1.2] that if X → X ′ is any closed embedding of schemesinducing an isomorphism on topological spaces, then (U → X ′) 7→ (U ×X′ X → X)is an equivalence of categories between the category of schemes étale over X ′ andthe category of schemes étale over X.

Remark 4. Lemma 2.1.1 holds for locally ringed spaces (same proof).

Proposition 2.1.2. A surjection of sheaves of rings f : A → B with stalkwisenilpotent kernel I ⊆ A is local in the sense that, for all a ∈ A, f(a) ∈ B∗ i a ∈ A∗.

Proof. The only diculty is to prove that, for a local section a ∈ A, the conditionf(a) ∈ B∗ implies a ∈ A∗. Being a unit can be checked on stalks, so we can reduceto the case of a surjection of rings A → B with nilpotent kernel I. If f(a) ∈ B∗,then since f is surjective, there is a a′ ∈ A such that f(a)f(a′) = 1. This impliesthat aa′ = 1 − i for some i ∈ I. But since I is nilpotent, 1 − i is invertible (withinverse 1+ i+ i2+ · · · ... the sum is nite since I is nilpotent), so a is invertible. Corollary 2.1.3. If X = (X,OX) is a locally ringed space, and f : OX′ → OX

is a surjection of sheaves of rings on X with stalkwise nilpotent kernel, then X ′ :=(X,OX′) is a locally ringed space (with the same underlying topological space as X)and (Id, f) : X → X ′ is a morphism of locally ringed spaces.

Proof. Apply the previous proposition to see that fx : OX′,x → OX,x is a local mapof local rings for each x ∈ X.

2.2. Square-zero thickenings. So that we can treat schemes and arbitrary ringedspaces on the same footing, we make the following

Denition 3. A map of ringed spaces f : X → X ′ is called a square-zero thickeningi it is an isomorphism on topological spaces and OX′ → OX is a square-zerothickening of sheaves of rings on that common space. (For locally ringed spaces orschemes it is equivalent, by Lemma 2.1.1, to ask only that f be a closed embeddingwith square zero ideal.)

Let f : A′ → A be a surjection of rings (or sheaves of rings) with square-zerokernel I. Then I has a natural A-module structure characterized by f(a′) · i = a′ifor any local sections a′ ∈ A′, i ∈ I. Indeed, this is just spelling out the A-modulestructure on I inherited from the natural isomorphisms

I ⊗A′ A = I/I2 = I.

Proposition 2.2.1. Let f : A′ → A be a surjection of rings with square-zero kernelI, so I has a natural A-module structure as described above. Set X := SpecA,


X ′ := SpecA′ so that f induces a map g := Spec f : X → X ′. Then g is anisomorphism on the level of topological spaces (which we suppress in the notation)and OX′ → OX is a surjective map of sheaves of rings on this space whose square-zero kernel (viewed as an OX-module as in the above discussion) is the quasi-coherentsheaf I∼ associated to the A-module I.

Proof. Since f is surjective, g is a closed embedding of schemes X → X ′ whoseideal sheaf J ⊆ OX′ is the quasi-coherent OX′-module associated to the A′-moduleI. That is, J = (I∼)′, where we put the ′ in the superscript to emphasize thatwe are referring to the functor from A′-modules to quasi-coherent OX′-modules, asopposed to the functor from A-modules to quasi-coherent OX-modules. This J issquare-zero since this can be checked on stalks and stalks of J are localizations of Iat primes of A′. In particular, g is an isomorphism on spaces by Lemma 2.1.1. Thenal statement follows from the computation

J ⊗OX′ OX = (I∼)′ ⊗OX′ OX

= (I ⊗A′ A)∼

= I∼

using [H, II.5.2e] for the second equality.

In fact, any square-zero thickening of the ane scheme X = SpecA with quasi-coherent kernel is of the form described in the above proposition:

Proposition 2.2.2. Let X = SpecA be an ane scheme,

0→ I → OX′ → OX → 0

a square-zero thickening of OX by an OX-module I. By Corollary 2.1.3, we have amap of locally ringed spaces (X,OX) → (X,OX′). Assume I is quasi-coherent sothat H1(X, I) = 0 and hence there is a square-zero thickening

0→ I(X)→ OX′(X)→ A→ 0

of the ring A by the A-module I(X). Set A′ := OX′(X). Then there is a naturalisomorphism X ′ = SpecA′ of locally ringed spaces under X = SpecA. In particular,(X,OX′) is an ane scheme.

Proof. By [EGA, Erratum I.1.8] (or see my paper [G, Prop. 7] for an easy proof), thelocally ringed space X ′′ := SpecA′ represents the presheaf on locally ringed spacesgiven by U 7→ HomAn(A

′,OU(U)), so the isomorphism A′ → OX′(X) yields a mapof locally ringed spaces X ′ → X ′′. This is a map of locally ringed spaces underX = SpecA because the diagram

A= // OX(X)

A′

OO

= // OX′(X)

OO


commutes by construction of A′ → A. By Proposition 2.2.1, X → X ′′ is a square-zero closed embedding of schemes (with ideal I(X)∼ = I), in particular it is anisomorphism on underlying spaces, so X ′ → X ′′ must also be an isomorphism onunderlying spaces and on this common space we have a commutative diagram withexact rows

0 // I

=

// OX′′

// OX

=

// 0

0 // I // OX′ // OX// 0

so OX′′ → OX′ has to be an isomorphism by the Five Lemma; i.e. X ′ → X ′′ is anisomorphism of locally ringed spaces under X.

Corollary 2.2.3. Let X = (X,OX) be a scheme, I a quasi-coherent OX-module,f : OX′ → OX a surjection of sheaves of rings on the space X with square zerokernel I. Then X ′ := (X,OX′) is a scheme, ane if X is ane, and hence (Id, f) :X → X ′ is a square-zero thickening of schemes.

Proof. The property of being a scheme is local, so we can work on an ane openpiece of X and reduce to Proposition 2.2.2.

2.3. Flatness.

Proposition 2.3.1. Let A′ → A be a surjection of rings (resp. sheaves of rings) withnilpotent (resp. stalkwise nilpotent) kernel I. For any A′-module M , the followingare equivalent:

(1) M is a at A′-module.

(2) TorA′

1 (M,A) = 0 and M ⊗A′ A = M/IM is a at A-module.

Proof. Certainly (1) implies (2) because atness is stable under base change. To seethat (2) implies (1), suppose M satises the two conditions. It is enough to checkatness on stalks and we have

TorA′

1 (M,A)x = TorA′

x1 (Mx, Ax)

(M ⊗A′ A)x = Mx ⊗A′xAx

at each point x, so the conditions on M are equivalent to the analogous conditionson each stalk of M . We thus reduce to the case where A′ → A is a surjection ofrings with nilpotent kernel; say In = 0. We must show that M is at over A′. It isequivalent to show that

TorA′

1 (M,N) = 0(2.3.1)

for every A′-module N . Given such an N , we have a nite ltration of N by A′-submodules

0 = InN ⊆ In−1N ⊆ · · · ⊆ IN ⊆ N.


Looking at the long exact sequence of Tors with M , we see that (2.3.1) would followfrom the vanishings

TorA′

1 (M, IkN/Ik+1N) = 0(2.3.2)

for k = 0, . . . , n− 1. The A′-modules IkN/Ik+1N are actually A-modules regardedas A′-modules by restriction of scalars along A′ → A (that is, they are annihilatedby I), so it is enough to prove the vanishing (2.3.1) when N is an A-module regardedas an A′-module via restriction of scalars. In this case, the functor _ ⊗A′ N factorsas the composition of the functor _ ⊗A′ A, followed by the functor _ ⊗A N . Weare trying to show that the rst left derived functor of _ ⊗A′N vanishes on M . Thehypothesis that TorA

′

1 (M,A) = 0 says that the rst left derived functor of _ ⊗A′ Avanishes on M and the hypothesis that M/IM is at over A ensures that the rstleft derived functor of _ ⊗A N vanishes on M ⊗A′ A = M/IM , hence we can usethe Grothendieck spectral sequence relating the left derived functors of our threeright exact functors to conclude the desired vanishing. In fact we make use only ofthe exact sequence of low-order terms

Tor2A(M ⊗A′ A,N)→ Tor1A′(M,A)⊗AN → Tor1A′(M,N)→ Tor1A(M ⊗A′ A,N)→ 0

and in any case, this spectral sequences argument could be replaced with somethingelementary (Exercise 2).

Remark 5. In the statement of Proposition 2.3.1, the condition TorA′

1 (M,A) = 0 isequivalent to saying that I⊗A′ M →M is injective, or equivalently, I⊗A′ M → IMis an isomorphism.

Remark 6. Proposition 2.3.1 appears as [H2, Prop. 2.2] (with the unnecessaryextra hypotheses that I2 = 0 and the rings are noetherian). I'm not sure whyhe put that noetherian hypothesis in there since he went through so much troublewhen proving [H2, Prop. 2.1] to introduce the basic idea of ltering a module andchecking Tor-vanishing on successive quotients, which is all one really needs to proveProposition 2.3.1.

Proposition 2.3.2. Consider a cartesian diagram of schemes

X //

f

X ′

f ′

Y // Y ′

where Y → Y ′ is a closed embedding with ideal I ⊆ OY ′. Then:

(1) X → X ′ is a closed embedding whose ideal J is the inverse image" of Iunder f ′ (the ideal of OX′ generated by the image of (f ′)♯ : (f ′)−1I → OX′,or, equivalently, the image of the natural map (f ′)∗I → OX′).

(2) If I is square-zero (resp. nilpotent, locally nilpotent, stalkwise nilpotent), thenso is J .


(3) If I is stalkwise nilpotent, (so X and X ′ have the same topological space bythe previous part and Lemma 2.1.1) then an OX′-module M is at over Y ′

i

0→ (f ′)∗I ⊗M →M → i∗M → 0

is exact (on the left) and i∗M is at over Y . In particular, f ′ is at i f isat and the natural map f ∗I → J is an isomorphism (i.e. is injective).

Proof. The rst statement is left to the reader as Exercise 3 and the second statementfollows from the rst because taking nth powers of an ideal commutes with takinginverse images. For the nal statement, rst note that Y and Y ′ have the sametopological space, as do X and X ′ and our cartesian diagram commutes, so thereis only one map of spaces involved here and we will just call it f : X → Y . Onthis space, we have a surjection f−1OY ′ → f−1OY of sheaves of rings with stalkwisenilpotent kernel f−1I. By Proposition 2.3.1, an OX′-module M is at over Y ′ (i.e. isat as an f−1OY ′-module) i 1) Torf

−1OY ′1 (M, f−1OY ) = 0 and 2) M⊗f−1OY ′ f

−1OY

is at over f−1OY . We just need to translate these two conditions onM into the twomore geometric looking conditions given in the nal statement of the proposition.

Condition (1) is equivalent to saying that the exact sequence

0→ f−1I → f−1OY ′ → f−1OY → 0

remains exact after tensoring over f−1OY ′ with M , i.e. that the sequence

0→ f−1I ⊗f−1OY ′ M →M → f−1OY ⊗f−1OY ′ M → 0(2.3.3)

is exact (the only issue is injectivity of the left map). We have

OX = OX′ ⊗f−1OY ′ f−1OY

(if this is not clear, see Exercise 3) and M is an OX′-module regarded as an f−1OY ′-module via restriction of scalars, so

f−1OY ⊗f−1OY ′ M = (f−1OY ⊗f−1OY ′ OX′)⊗OX′ M

= OX ⊗OX′ M

= i∗M

and similarly

f−1I ⊗f−1OY ′ M = (f−1OY ⊗f−1OY ′ OX′)⊗OX′ M

= f ∗I ⊗OX′ M,

so the sequence (2.3.3) is just the sequence appearing in the nal statement of theproposition.

As for condition (2), we already noted that M ⊗f−1OY ′ f−1OY = i∗M , so the con-

dition that this is at over f−1OY is just the second condition in the nal statementof the proposition.


2.4. Descent of properties of morphisms. In this section we will be concernedwith the following sort of questions: Suppose

Xi //

f

X ′

f ′

Y

j // Y ′

(2.4.1)

is a commutative diagram of schemes where i and j are some kind of nilpotentthickening" as in Denition 2. Suppose f has some property. Does this imply f ′

has the same property? Here are some typical results along these lines:

Lemma 2.4.1. Consider a commutative diagram of schemes (2.4.1).

(1) Assume i and j are homeomorphisms (on underlying topological spaces).Then f is open (resp. closed, quasi-compact) i f ′ is open (resp. closed,quasi-compact).

(2) Assume i is a universal homeomorphism and j is a monomorphism (e.g. aclosed embedding). Then f is separated i f ′ is separated.

(3) Assume i and j are universal homeomorphisms. Then f is universally open(resp. universally closed, . . . ) i f ′ is universally open (resp. universallyclosed, . . . ).

Proof. The statement (1) is an elementrary exercise in topology since the propertiesin question are all properties of the underlying maps of topological spaces (e.g.according to [EGA, IV.1.1], a map of schemes f : X → Y is quasi-compact if−1(U) is quasi-compact for each quasi-compact open subspace U ⊆ Y ).

For (2), we rst note that a map of schemes f : X → Y is separated i ∆ = ∆f :X → X ×Y X is a closed embedding on the level of topological spaces (for then it isautomatically a closed embedding of schemes since the stalk of ∆−1OX×Y X → OX

at x ∈ X is always surjective because it is the localization of the the surjectivemultiplication map OX,x ⊗OY,f(x)

OX,x → OX,x at the preimage of mx). Thus wereduce to showing that the map (i, i) in the commutative diagram

Xi //

∆f

X ′

∆f ′

X ×Y X(i,i)

// X ′ ×Y ′ X ′

is a homeomorphism under the indicated hypotheses. First of all, X×Y X = X×Y ′Xbecause we assume j is a monomorphism, so we want to show that (i, i) : X×Y ′X →X ′ ×Y ′ X ′ is a homeomorphism. In fact it is a universal homeomorphism becauseuniversal homeomorphisms are closed under composition and base change and (i, i)is a composition of (Id, i) and (i, Id) (see below) and each of the latter maps is a


base change of the universal homeomorphism i as in the cartesian diagrams:

X ×Y ′ X(i,Id)

//

π1

X ′ ×Y ′ X

π1

X

i // X ′

X ′ ×Y ′ X(Id,i)

//

π2

X ′ ×Y ′ X ′

π2

X

i // X ′

For (3) we rst treat the implication =⇒ . Consider an arbitrary map V ′ → Y ′.We want to show that π2 : X ′ ×Y ′ V ′ → V ′ is open (resp. closed) assuming f isuniversally open (resp. closed). We have a commutative diagram

X ×Y ′ V ′(i,Id)

//

(f,Id)

X ′ ×Y ′ V ′

π2

Y ×Y ′ V ′

p2 // V ′

where (f, Id) is open (resp. closed) because it is a base change of f and (i, Id) (resp.p2) is a homeomorphism because it is a base change of i (resp. j), hence π2 is open(resp. closed) by (1).

For the converse in (3), consider an arbitrary map U → Y . We want to showπ1 : U ×Y X → U is open (resp. closed) assuming f ′ is universally open (resp.closed). We have a commutative diagram with cartesian squares

U ×Y X //

q

X

η

i

&&MMMMM

MMMMMM

MM

U ×Y ′ X ′ //

p

Y ×Y ′ X ′p2 //

p1

X ′

f ′

U // Y

j // Y ′

where η is the map induced by the commutative diagram (2.4.1), so p1η = f , andπ1 = pq. The map p2 is a universal homeomorphism because it is a base change ofj, hence η must also be a universal homeomorphism by two-out-of-three" becausei is a universal homeomorphism. Thus we see that q is a homeomorphism, so ourmap π1 is open (resp. closed) because it is a composition of q and the map p, whichis a base change of f ′.

Lemma 2.4.2. Consider a pushout diagram of rings

Af // B

A′

i

OO

f ′// B′

j

OO


where i (hence also j) is surjective with square-zero kernel I. If f is nite type, thenf ′ is nite type. If f is nitely presented and

TorA′

1 (I, B′) = 0(2.4.2)

TorA′

1 (A,B′) = 0

(these conditions hold, for example, if f ′ is at), then f ′ is nitely presented.

Remark 7. The converses of both statements of course hold without any assump-tions on i, j since nite type (resp. presentation) maps are stable under pushout.

Proof. Since the square is a pushout, the kernel J of the surjection j is given byJ = f ′(I)B. Suppose f is nite type. Choose b1, . . . , bm ∈ B generating B as anA-algebra. Choose any lifts b′1, . . . , b

′m ∈ B′ of the bi, so j(b′i) = bi.

I claim the b′i generate B′ as an A′-algebra. Given b′ ∈ B′, we can write

j(b′) =∑I

f(aI)bI

for some nite set of exponents I ∈ Nm and some aI ∈ A. (We write bIinstead of

bi11 · · · bimm when I = (i1, . . . , im).) Choose lifts a′I ∈ A′ of the aI . Then∑

I f′(a′I)(b

′)I

has the same image under j as b′, so we can write

b′ =∑I

f ′(a′I)(b′)I + f ′(a′)b′′(2.4.3)

for some a′ ∈ I ⊆ A′ and some b′′ ∈ B′. Repeating the same argument with b′"replaced by b′′," we can write

b′′ =∑J

f ′(a′′J)(b′)J + f ′(a′′)b′′′(2.4.4)

for a nite set of exponent vectors J and some a′′J ∈ A′, a′′ ∈ I, b′′′ ∈ B′. Since

f ′(a′)f ′(a′′) = f ′(a′a′′) = f(0) = 0

because I2 = 0, equations (2.4.3) and (2.4.4) yield

b′ =∑I

f ′(a′I)(b′)I +

∑J

f ′(a′a′′J)(b′)J ,

which shows that our arbitrary element b′ ∈ B′ is in the A′-subalgebra generated bythe b′i, as desired, so the map

A′[x] → B′(2.4.5)

g(x) 7→ g(b′)

is surjective. (We write x for x1, . . . , xm.)

Let L′ ⊆ A′[x] be the kernel of (2.4.5) and let L ⊆ A[x] be the kernel of

A[x] → B(2.4.6)

g(x) → g(b).


It remains to prove that L′ is a nitely generated ideal in A′[x] when L is a nitelygenerated ideal in A[x] and we have (2.4.2). The assumptions (2.4.2) ensure thatI ⊗A′ B′ = J and that the sequence

0→ L′ → A′[x]→ B′ → 0

remains exact after I ⊗A′ _ , so we have an exact sequence

0→ I ⊗A′ L′ → I[x]→ J → 0(2.4.7)

where I[x] = I⊗A′ A′[x] is the A′[x]-module of polynomials in x with coecients inI". The left map in (2.4.7) is a′ ⊗ g(x) 7→ a′g(x) and the right map is g(x) 7→ g(b

′).

Choose generators l1(x), . . . , ln(x) for L. Choose lifts l′′1(x), . . . , l′′n(x) ∈ A′[x] of

the li(x). Then, for i = 1, . . . , n, l′′i (b′) ∈ J since j(l′′i (b

′)) = li(b) = 0, so we can

write l′′i (b′) = f(a′i)b

′′i for some a′i ∈ I ⊆ A′ and some b′′i ∈ B′. Since (2.4.5) is

surjective, we can write b′′i = gi(b′) for some gi(x) ∈ A′[x]. Then

l′i(x) := l′′i (x)− a′igi(x)

is in L′ and lifts li(x). (The primes on polynomials in this proof have nothing to dowith dierentiation.)

I claim that l′1(x), . . . , l′n(x) ∈ L′ generate L′. Given an arbitrary l′(x) ∈ L′, the

image l(x) ∈ A[x] is in L, so we can write

l(x) =n∑

i=1

gi(x)li(x)(2.4.8)

for some gi(x) ∈ A[x]. Choose lifts g′i(x) ∈ A′[x] of the gi(x). Then∑

i g′i(x)l

′i(x)

have the same image in A[x], so, since

I[x] = Ker(A′[x]→ A[x]),

we can write

l′(x) =n∑

i=1

g′i(x)l′i(x) + t(x)(2.4.9)

for some t(x) ∈ I[x]. Evaluating (2.4.9) at b′then shows that t(x) is in the kernel

of the right map in (2.4.7), so we can write

t(x) = a′l′′(x)(2.4.10)

for some a′ ∈ I ⊆ A′ and some l′′(x) ∈ L′. Repeating the same argument with l′(x)replaced by l′′(x), we can write

l′′(x) =n∑

i=1

hi(x)l′i(x) + s(x)(2.4.11)


for some hi(x) ∈ A′[x] and some s(x) ∈ I[x]. Since I2 = 0, we have a′s(x) = 0,hence (2.4.9) and (2.4.10) yield

l′(x) =n∑

i=1

(g′i(x) + a′hi(x))l′i(x),

as desired.

Remark 8. The assumptions (2.4.2) in Lemma 2.4.2 cannot be removed. For exam-ple, take i = f ′ any surjection with square zero kernel which is not nitely presented(e.g. A = B′ = Z, A′ = Z[x1, x2, . . . ]/(xixj), f ′ : A′ → Z killing all the xi). Thenthe map f obtained by pushing out is even an isomorphism, but we can't concludenite presentation of f ′.

Lemma 2.4.3. Consider a commutative diagram of schemes (2.4.1).

(1) If i and j are stalkwise-nilpotent thickenings, then f is open (resp. closed,quasi-compact, universally open, universally closed, separated) i f ′ is open(resp. closed, . . . ).

(2) If the diagram is cartesian, and j (hence i) is a square zero thickening, thenf ′ is of locally nite type i f is of locally nite type.

(3) If the diagram is cartesian, j (hence i) is a square zero thickening, and f ′ isat, then f ′ is of locally nite presentation (resp. nite presentation, proper,...) i f is of locally nite presentation (resp. nite presentation, ...).

Proof. (1) follows from Lemma 2.4.1 since a stalkwise-nilpotent thickening is a closedembedding which is a universal homeomorphism on spaces (Lemma 2.1.1). (2) andthe rst part of (3) follow from Lemma 2.4.2 by working locally. The other parts of(3) are obtained by combining the rst part of (3) with (1) (for example, proper"equals nitely presented, separated, and universally closed," [EGA, II.5.4.1] andnitely presented" equals locally nitely presented and quasi-compact").

2.5. Descent of properties of modules. One can also consider the followingvariant of the question we looked at in the last section: Suppose i : X → X ′ is asquare-zero thickening (or some other kind of nilpotent thickening) of schemes andM ′ is an OX′-module. What properties of the OX-module M := i∗M ′ are inheritedby M ′?

Lemma 2.5.1. Let A′ → A be a square zero surjection of rings with kernel I, M ′

an A′-module. Suppose that the reduction M := M ′/IM ′ is a nitely presentedA-module and the natural sequence

0→ I ⊗A M →M ′ →M → 0

is exact (the left map is i ⊗ m 7→ im′, where m′ ∈ M ′ is any lift of m ∈ M , thechoice being irrelevant since I2 = 0). Then M ′ is a nitely presented A′-module.


Proof. Pick a presentation Am → An → M → 0 of M . The result will follow froma diagram chase once we construct a commutative diagram

0 0 0

Am //

OO

An //

OO

M //

OO

0

(A′)m //

OO

(A′)n //

OO

M ′

OO

// 0

Im //

OO

In

OO

// I ⊗A M

OO

// 0

0

OO

0

OO

0

OO

of A′-modules where the columns and the top and bottom row are exact (the diagramchase then shows that the middle row is exact). To construct this diagram, we rstconstruct the upper right square by choosing lifts m′i ∈ M ′ of the images mi ∈ Mof the standard basis vectors ei ∈ An under An → M , then we dene (A′)n → M ′

by taking the standard basis vector e′i to m′i. Since (A′)n → An is also surjective,we can construct the upper left square in the same way. The bottom row is thenthe top row tensored over A with I (which is exact by right exactness of tensorproduct) and the vertical arrows from the bottom row to the middle row are thenatural maps as in the statement of the lemma; it is key that the right column isexact by hypothesis. Lemma 2.5.2. Let i : X → X ′ be a square-zero thickening of schemes with idealI and let M ′ be an OX′-module. Set M := i∗M ′ = M ′/IM ′. If M and IM ′ arequasi-coherent OX-modules, then M ′ is a quasi-coherent OX′-module. If the naturalsequence

0→ I ⊗M →M ′ →M → 0

is exact and M is a quasi-coherent OX-module of locally nite presentation, thenM ′ is a quasi-coherent OX′-module of locally nite presentation.

Proof. When M and IM ′ are quasi-coherent OX-modules, they are also quasi-coherent as OX′-modules (via restriction of scalars) (the pushforward of a quasi-coherent sheaf under a closed embedding is always quasi-coherent), so the rststatement reduces to the (standard) assertion that an extension of quasi-coherentsheaves is quasi-coherent. This assertion is local so we can assume X ′ is ane andthe result follows from the Five Lemma in the diagram of OX′-modules

0 // Γ(X ′, IM ′)∼

∼=

// Γ(X ′,M ′)∼

// Γ(X,M)∼

∼=

// 0

0 // IM ′ // M ′ // M // 0


(the top row is exact since H1(X ′, IM ′) = 0 because X ′ is ane and IM ′ is quasi-coherent).

When the natural sequence is exact (i.e. IM ′ = I ⊗M) and M is quasi-coherent,M ′ is quasi-coherent by the rst part and the niteness statement then boils downto the previous lemma since it is local in nature.

3. Differentials and Algebra Extensions

In this section we recall some basic facts about Kähler dierentials and algebraextensions (c.f. [Ill, II.1], [EGA, 0, 20]).

3.1. Denitions of dierentials. Let f : A → B be a ring homomorphism. Themodule of (Kähler) dierentials ΩB/A is the B-module generated by symbolds db,b ∈ B, subject to the relations d(f(a)) = 0 for all a ∈ A and

d(b1b2) = b1db2 + b2db1(3.1.1)

for all b1, b2 ∈ B. We will usually suppress f in the notation and just write therst relations da = 0. The second relation is called the Leibnitz Rule and the rstrelation is called A-linearity. The two relations in particular imply that d(ab) = adbfor a ∈ A, b ∈ B, so d : B → ΩB/A is an A-module map (regarding the B-moduleΩB/A as an A-module by restriction of scalars along f).

It is clear from this denition that Ω_ /_ commutes with ltered direct limits inthe sense that if (Ai → Bi) is a ltered direct limit system of maps of rings withdirect limit A→ B, then ΩB/A = lim

−→ΩBi/Ai

. (We will see later that Ω_ /_ in fact

commutes with all direct limits" in an appropriate sense.)

Alternatively, let I = IB/A be the kernel of the multiplication map B ⊗A B → Bgiven by b1 ⊗ b2 7→ b1b2, so we have an exact sequence

0→ I → B ⊗A B → B → 0

of B ⊗A B-modules and hence also an exact sequence

0→ I/I2 → (B ⊗A B)/I2 → B → 0

where (B ⊗A B)/I2 → B is now a surjection of rings with square-zero kernel I/I2,hence I/I2 has a natural B-module structure (2). For b ∈ B and i ∈ I/I2, thescalar product b · i ∈ I/I2 is represented by both the class of (b⊗ 1)i and the classof (1⊗ b)i in I/I2 (because both b⊗ 1 and 1⊗ b are lifts of b to (B ⊗A B)/I2).

In fact, one could alternatively note right from the beginning that B ⊗A B → Bhas two obvious sections (namely b 7→ b ⊗ 1 and b 7→ 1 ⊗ b), so we could pickone of those sections once and for all and use it to regard any B ⊗A B-module asa B-module, but it is very useful to note that as long as that B ⊗A B-module isannihilated by I, it doesn't matter which of the two sections we use to regard it asa B-module.


Proposition 3.1.1. The map

ΩB/A → I/I2

db 7→ b⊗ 1− 1⊗ b

denes an isomorphism of B-modules ΩB/A → I/I2 with inverse

I/I2 → ΩB/A∑i

b1,i ⊗ b2,i 7→∑i

b2,idb1,i.

Proof. We rst check that the map ΩB/A → I/I2 is well-dened (i.e. that it killsthe relations on the symbols db used to dene ΩB/A). It clearly kills da for a ∈ Abecause a⊗ 1− 1⊗ a is zero in B ⊗A B. To see that it respects the Leibnitz Rule,we compute

b · (c⊗ 1− 1⊗ c) + c · (b⊗ 1− 1⊗ b)

= (b⊗ 1)(c⊗ 1− 1⊗ c) + (1⊗ c)(b⊗ 1− 1⊗ b)

= (bc)⊗ 1− b⊗ c+ b⊗ c− 1⊗ (bc)

= (bc)⊗ 1− 1⊗ (bc).

We next check that the map I/I2 → ΩB/A is well-dened. We rst note that(b1, b2) 7→ b2db1 is an A-bilinear map B × B → ΩB/A (because d : B → ΩB/A is A-linear), so our formula at least denes a morphism of A-modules B ⊗A B → ΩB/A.Let us regard B ⊗A B as a B-module via the ring homomorphism b 7→ 1⊗ b. Thenin fact it is clear that our A-linear map is B-linear. Restricting to I ⊆ B ⊗A B, weobtain a B-linear map I → ΩB/A. (If we had chosen to regard B⊗AB as a B moduleusing the other section b 7→ b ⊗ 1, then our A-linear map B ⊗A B → ΩB/A wouldnot be B-linear, but its restriction to I ⊆ B⊗AB would still be B-linear.) We nextclaim that our B-module map I → ΩB/A kills I2. Indeed, suppose

∑i b1,i ⊗ b2,i and∑

j c1,j ⊗ c2,j are both in I, so∑

i b1,ib2,i = 0 and∑

j c1,jc2,j = 0. We need to showthat our map kills their product

∑i,j(b1,ic1,j)⊗ (b2,ic2,j). For this we compute (using

the Leibnitz Rule)∑i,j

b2,ic2,jd(b1,ic1,j) =∑i,j

b1,ib2,ic2,jdc1,j + b2,ic1,jc2,jdb1,i

= 0.

(The rst term vanishes because for each xed j the sum over i vanishes and thesecond term vanishes because for each xed i the sum over j vanishes.)

We have shown that our maps are well-dened. It remains only to check thatthe compositions are identities. To see that ΩB/A → ΩB/A is the identity, we justcompute

db 7→ b⊗ 1− 1⊗ b

7→ 1db− bd1

= db.


To see that I/I2 → I/I2 is the identity we compute∑i

b1,i ⊗ b2,i 7→∑i

b2,idb1,i

7→∑i

b2,i · (b1,i ⊗ 1− 1⊗ b1,i)

=∑i

(1⊗ b2,i)(b1,i ⊗ 1− 1⊗ b1,i)

=∑i

b1,i ⊗ b2,i − 1⊗ (∑i

b1,ib2,i)

=∑i

b1,ib2,i.

Remark 9. The above proof of Proposition 3.1.1 is not very enlightened," but itis straightforward. The isomorphim of that proposition is clearly natural in A→ B.

3.2. Sheacation. Let FlAn be the category whose objects are maps of rings andwhose maps are commutative squares. Let AnMod be the category whose objectsare pairs (A,M) consisting of a ring A and an A-module M , where a morphism(A,M)→ (B,N) is a pair (f, g) consisting of a ring map f : A→ B and an f -linearmap g : M → N (equivalently, a map M ⊗f

A B → N of B modules). Then themodule of dierentials denes a functor

Ω_ /_ : FlAn → AnMod

(A→ B) 7→ (B,ΩB/A).

This functor takes a commutative diagram

A //

A′

B

f // B′

of rings (a FlAn-morphism (A → B) → (A′, B′)) to the AnMod map (f, df) :(B,ΩB/A)→ (B′,ΩB′/A′) where df(db) := d(f(b)).

We can sheafy the construction of Kähler dierentials as follows. Suppose A→ Bis a map of sheaves of rings on a topological space X. Then the above functorialityof Ω_ /_ endows the presheaf

ΩpreB/A : U 7→ ΩB(U)/A(U)

with the structure of a presheaf of B-modules, so we can obtain a B-module ΩB/A

by taking the sheaf associated to this presheaf. It is clear from this denition andthe ltered-direct-limit-preservation of Ω_ /_ that the stalks of ΩB/A (which arethe same as the stalks of Ωpre

B/A) are given by

(ΩB/A)x = ΩBx/Ax .(3.2.1)


The same argument shows, more generally, that for a map of sheaves of rings A→ Bon a space Y and a map of spaces f : X → Y , we have

Ωf−1B/f−1A = f−1ΩB/A.(3.2.2)

Indeed, one can rst check this using the presheaf variants of Ω and f−1where itfollows from preservation of ltered direct limitsthen obtain the indicated identityby sheafying.

Alternatively, let I be the kernel of B ⊗A B → B and set ΩB/A := I/I2 as inthe usual case of rings. To see that this gives you the same B-module as the aboveconstruction of ΩB/A, you can rst perform all of these constructions as presheaves(i.e. take the presheaf tensor product B⊗AB, let J be the kernel of the multiplicationmap, and let J/J2 denote the presheaf quotient). Since the presheaf tensor productsheaes to the usual tensor product, J sheaes to I and J/J2 sheaes to I/I2

(sheacation is exact). On the other hand J/J2 is isomorphic to the presheaf ΩpreB/A

dened above because J(U)/J2(U) is isomorphic to ΩpreB/A(U) naturally in the open

subspace U ⊆ X by Proposition 3.1.1, so if we sheafy this isomorphism, we getan isomorphism between I/I2 and the B-module ΩB/A dened as in the previousparagraph.

The formation of ΩB/A is again natural in the map A→ B of sheaves of rings onX. Let FlAn(X) be the category of maps of sheaves of rings onX, where morphismsare commutative squares. Let AnMod(X) be the category whose objects are pairs(A,M) consisting of a sheaf of rings A ∈ An(X) and an A module M . A morphism(f, g) : (A,M)→ (B,N) in AnMod(X) is a pair consisting of a map f : A→ B inAn(X) and an f linear map g : M → N (equivalently, a map M ⊗f

A B → N of Bmodules). Then Ω_ /_ denes a functor

Ω_ /_ : FlAn(X) → AnMod(X).(3.2.3)

3.3. Algebra extensions. Let A → B be a homomorphism of rings or sheaves ofrings, M a B-module. For us, an A-algebra extension of B by M is a surjectionof A-algebras B′ → B with square-zero kernel identied with M as a B-module.(Recall from 2.2 that the kernel of a square zero surjection B′ → B has a naturalB-module structure.) Such an algebra extension can be viewed as an A-algebra B′

together with an exact sequence of B′-modules

0→M → B′ → B → 0

where B′ → B is a map of A-algebras and M ∈ Mod(B) is regarded as a B′-module by restriction of scalars along B′ → B. If there is no risk of confusion, suchan algebra extension will be denoted simply by B′. These algebra extensions forma category where a morphism is a commutative diagram

0 // M // B′′

// B // 0

0 // M // B′ // B // 0


where B′′ → B′ is a map of A-algebras. The Five Lemma implies that every suchmorphism is an isomorphism. The set of isomorphism of classes of A-alegbra exten-sions of B by M is denoted ExalA(B,M).

The set ExalA(B,M) has a natural B-module structure. The underlying abeliangroup structure is dened using the usual Baer sum" of extensions:

B′ +B′′ := B′ ×B B′′/(m, 0)− (0,m) : m ∈M.For b ∈ B and B′ ∈ ExalA(B,M), the scalar multiple b · B′ ∈ ExalA(B,M) is theA-algebra extension of B by M ′ in the bottom row of

0 // M

b·_

// B′

// B // 0

0 // M // b ·B′ // B // 0

where b · B′ is dened (up to isomorphism) by demanding that the left square bea pushout, and the map b · B′ → B should be the map obtained from the mapsB′ → B and 0 : M → B via the universal property of pushout.

The zero element in the B-module ExalA(B,M) is the trivial square zero extensionof B by M , denoted B[M ]. As an additive abelian group, B[M ] := B ⊕M . Themultiplication in B[M ] is given by

(b,m)(b′,m′) := (bb′, bm′ + b′m)

for local sections b, b′ of B and m,m′ of M , and the A-algebra structure on B[M ] isgiven by a 7→ (a, 0), supressing notation for A→ B. The map (b,m)→ b is clearly asurjective A-algebra homomorphism with square-zero kernel M = (0,m) ∈ B[M ].This surjection has an A-algebra section b 7→ (b, 0). In fact, an algebra extensionB′ of B by M is trivial (isomorphic to the trivial extension) i B′ → B has an A-algebra section. Indeed, given such a section s : B → B′, we obtain an isomorphismfrom the trivial extension B[M ] to B′ using the map (b,m) 7→ s(b) +m.

One checks that the above notions of addition and scalar multiplication makeExalA(B,M) into a B-module in the same way one checks that the Yoneda descrip-tion of Ext1B(M,N) yields a B-module. For example, let us check that 0 · B′ is(isomorphic to) the trivial algebra extension for any B′ ∈ ExalA(B,M). By the def-inition of scalar multiplication of A-algebra extensions this is equivalent to showingthat there is a map of A-algebra extensions as below.

0 // M

0

// B′g //

B // 0

0 // M // B[M ] // B // 0

Indeed, b′ 7→ (g(b′), 0) is such a map.

Remark 10. The annihilator of the B-module M is contained in the annihilator ofthe B-module ExalA(B,M) (for any ring map A→ B). Indeed, if b ∈ AnnM , thenb ·B′ is the trivial extension for any B′ ∈ ExalA(B,M) because b ·B′ is obtained by


pushing out B′ along b · _ : M →M , which is the zero map when b ∈ AnnM , andwe just saw above that pushing out along the zero map yields the trivial extension.

Theorem 3.3.1. Let A → B be a surjection of rings (or sheaves of rings) withkernel I. Then for any B-module M , there is a natural isomorphism of B-modules

ExalA(B,M) = HomMod(B)(I/I2,M).

When M = I/I2, this isomorphism takes the rst innitesimal neighborhood A/I2 ∈ExalA(B, I/I2) to Id ∈ HomMod(B)(I/I

2, I/I2).

Proof. Constructing the map ExalA(B,M) → HomMod(B)(I/I2,M) is easy: Given

an algebra extension

0 // M // B′ // B // 0

Aj

`ÀAAAAAAAf

OO

and an i ∈ I = Ker f , commutativity implies j(i) ∈ Ker g = M . For i1, i2 ∈I, j(i1i2) = j(i1)j(i2) = 0 since both j(i1) and j(i2) are in M and the latter issquare zero in C. Thus we get a map I/I2 → M which is easily seen to be B-linear (everything is inherited from B′). It is obvious that this map takes the rstinnitesimal neighborhood to the identity map Id ∈ HomMod(B)(I/I

2, I/I2).

To construct an inverse

HomB(I/I2,M)→ ExalA(B,M),

pushout the rst innitesimal neighboorhood exact sequence (note B = A/I)

0 // I/I2 // A/I2 // B // 0

along I/I2 → M . It is also obvious that this takes Id ∈ HomB(I/I2, I/I2) to the

rst innitesimal neighborhood extension! Checking that these two maps are inverseand are maps of B-modules will be left to the reader (Exercise 4).

The reader may also wish to read the rst part of 7.3, particularly Proposi-tion 7.3.1, for another interesting example where one can explicitly compute theB-module ExalA(B,M) by hand."

3.4. Adjointness property. The functor Ω_ /_ (see (3.2.3)) has a right adjoint.For (A,M) ∈ AnMod(X), let A[M ] denote the trivial square zero extension of Aby M (3.3) and let A → A[M ] be the natural map of sheaves of rings on X givenby a 7→ (a, 0). Formation of A→ A[M ] determines a functor

[_ ] : AnMod(X) → FlAn(X)

(A,M) 7→ (A→ A[M ])


which is easily seen to be right adjoint to Ω:

HomAnMod(X)((B,ΩB/A), (C,M)) = HomFlAn(X)((f : A→ B), (C → C[M ]))

(g, k) 7→ Af //

gf

B

g×(b7→k(db))

C // C[M ]

(g, db 7→ D(b)) ← [ Af //

gf

B

g×D

C // C[M ]

(It is completely straightforward to check that the maps here are well-dened andinverse to each other.)

In particular, taking B = C, g = Id above, we nd that HomMod(B)(ΩB/A,M) isbijective with the set of A-algebra morphisms B → B[M ] of the form Id×D. If fis an epimorphism1 in An(X), then for any M the only possibility is D = 0, so wend ΩB/A = 0.

3.5. Derivations. In general, a function D : B → M yields an A-algebra mapId×D : B → B[M ] i D is A-linear and satises

D(b1b2) = b1D(b2) + b2D(b1)(3.5.1)

for all b1, b2 ∈ B. Such a map D is called an A-linear derivation from B to M . Theset of such A-linear derivations is denoted DerA(B,M). As mentioned above, thespecial case of the adjointness relation of 3.4 where B = C, g = Id yields a naturalbijection

HomMod(B)(ΩB/A,M) = DerA(B,M)(3.5.2)f 7→ (b 7→ f(db))

(db 7→ D(b)) ←[ Dfor all B-modules M , which characterizes ΩB/A up to unique isomorphism. Thebijection (3.5.2) identies the B-module structure on HomMod(B)(ΩB/A,M) with theB-module structure on DerA(B,M) where b ·D is dened by (b ·D)(b′) := b ·D(b′).The isomorphism (3.5.2) is natural in A → B in the sense that for a commutative

1We always use epimorphism" in the categorical sense. A surjection in An(X) is an An(X)morphism f : A → B where f is an epimorphism of sheaves on X (surjective on stalks). Asurjection in An(X) is also an epimorphism in An(X), but not conversely: localizations andcompletions are epimorphisms in An(X) but not generally surjections.


diagram of rings (or sheaves of rings)

A //

A′

B

g // B′

and a B′-module M , the diagram

HomMod(B)(ΩB/A,M) // DerA(B,M)

HomMod(B′)(ΩB′/A′ ,M)

OO

// DerA′(B′,M)

g∗

OO

commutes, where the horizontal arrows are the natural bijections (3.5.2) and theleft vertical arrow is induced by the g-linear map dg : ΩB/A → ΩB′/A′ (3.2). Notethat M is being regarded as a B-module by restriction of scalars along g, so

HomMod(B)(ΩB/A,M) = HomMod(B′)(ΩB/A ⊗B B′,M).

Lemma 3.5.1. Let A→ B be a map of rings or sheaves of rings. The map takingan A-linear derivation D : B →M to the diagram

0 // M // B[M ]

b 7→(b,D(b))

// B // 0

0 // M // B[M ] // B // 0

establishes an isomorphism of groups between DerA(B,M) and the group of auto-morphisms of the trivial square-zero extension B[M ] in the category of A-algebraextensions of B by M (3.3).

Proof. This is a straightforward exercise and also a special case of Lemma 5.1.3 (takeT = B, T ′ = B[M ] there), which we will come to later.

3.6. Direct limit preservation. Since Ω is a left adjoint, it preserves direct limitsin FlAn(X): If (fi : Ai → Bi)i is a direct limit system in FlAn(X) with limitf : A → B (that is, A is the limit of the Ai, B is the limit of the Bi, and f is thelimit of the fi), then the natural map

lim−→

ΩBi/Ai⊗Bi

B → ΩB/A(3.6.1)

is an isomorphism of B modules. In particular, if B = B1⊗AB2 is a tensor product,we have:

ΩB/A = (ΩB1/A ⊗B1 B)⊕ (ΩB2/A ⊗B2 B)(3.6.2)ΩB/B2 = ΩB1/A ⊗B1 B.(3.6.3)


3.7. Exact sequences. The purpose of this section is to use our basic theory ofalgebra extensions to derive various exact sequences relating modules of dierentialsfor dierent ring maps.

Theorem 3.7.1. Let f : A→ B, g : B → C be maps of rings (or sheaves of rings).Then for any C-module M , there is a natural exact sequence of C-modules

0 // DerB(C,M) // DerA(C,M) // DerA(B,M)δ //

δ // ExalB(C,M) // ExalA(C,M) // ExalA(B,M)

where the map δ takes an A-linear derivation D : B →M to the (isomorphism classof the) B-algebra extension of C by M in the diagram below:

0 // M // C[M ] // C // 0

B

g

OO

g×D

b bEEEEEEEE

Proof. There are ve places where one has to check exactness. We will leave four ofthem as exercises for the reader (Exercise 5) and just check exactness at ExalB(C,M).To say that C ′ ∈ ExalB(C,M) maps to zero (the trivial extension) in ExalA(C,M)is to say that the B-algebra surjection C ′ → C has a section s : C → C ′ which isa map of A-algebras when C ′ and C are regarded as A-algebras by precomposingtheir B-algebra structures (g′ : B → C ′ and g : B → C) with f : A → B. Thatis: g′f = sgf . Using the section s we obtain an A-linear derivation D : B → M bysetting D(b) := g′(b)− sg(b). Note that Df = 0 because g′f = sgf and D satisesthe Leibnitz Rule (3.5.1) because

D(b1b2) = g′(b1b2)− sg(b1b2)

= g′(b1)g′(b2)− sg(b1)sg(b2)

= g′(b1)(g′(b2)− sg(b2)) + sg(b2)(g

′(b1)− sg(b1))

= b1 ·D(b2) + b2 ·D(b1),

where, in the last equality we used the fact that g′(b1) lifts g(b1) ∈ C to C ′ andsg(b2) lifts g(b2) ∈ C to C ′ (M is regarded here as a B-module by restriction ofscalars along g). We claim that C ′ is the image of D under δ. To see this, we needonly check that the map

h : C[M ] → C ′

(c,m) 7→ s(c) +m

is a morphism of B-algebras when C[M ] is regarded as a B-algebra via g × D :B → C[M ] (because h obviously commutes with the maps to C and from M , so itwill then be a morphism of B-algebra extensions of C by M , hence necessarily an


isomorphism (3.3)). To see that h is a ring homomorphism we compute

h((c1,m1)(c2,m2)) = h(c1c2, c1 ·m2 + c2 ·m1)

= s(c1c2) + c1 ·m2 + c2 ·m1

= (s(c1) +m1)(s(c2) +m2)

= h(c1,m1)h(c2,m2).

We used the fact that M is square-zero in the third equality to know that m1m2 = 0in C ′ and ci ·mj = s(ci)mj because s(ci) lifts ci ∈ C to C ′. It remains only to checkthat h commutes with the maps from B, which is easy:

h(g ×D)(b) = h(g(b), g′(b)− sg(b))

= sg(b) + g′(b)− sg(b)

= g′(b).

Corollary 3.7.2. Let f : A→ B, g : B → C be maps of rings (or sheaves of rings).Then there is an exact sequence of C-modules

ΩB/A ⊗B C → ΩC/A → ΩC/B → 0(3.7.1)

and the following are equivalent:

(1) ΩB/A ⊗B C → ΩC/A has a retract (i.e. the left map is also injective and theresulting short exact sequence splits).

(2) ExalB(C,M)→ ExalA(C,M) is injective for all C-modules M .(3) ExalB(C,ΩB/A ⊗B C)→ ExalA(C,ΩB/A ⊗B C) is injective.(4) The B-algebra extension of C by ΩB/A ⊗B C below is trivial:

0 // ΩB/A ⊗B C // C[ΩB/A ⊗B C] // C // 0

B

g

OO

b7→(g(b),db⊗1)

ffMMMMMMMMMMM

Proof. For the rst statement: For any C-module M , the sequence of C-modulesobtained by applying HomMod(C)(_ ,M) to the sequence (3.7.1) is identied, viathe isomorphism (3.5.2) of 3.5, with the sequence

0→ DerB(C,M)→ DerA(C,M)→ DerA(B,M).(3.7.2)

This latter sequence is exact by the theorem and since this is true for any M , itfollows formally (Exercise 6) that the sequence (3.7.1) is exact.

For the second part, Theorem 3.7.1 says that (3.7.2) is actually part of a longerexact sequence

0 // DerB(C,M) // DerA(C,M) // DerA(B,M)δ //

δ // ExalB(C,M) // ExalA(C,M) // ExalA(B,M)

(3.7.3)


For (1) implies (2): If g has a retract, then 0 → ΩB/A ⊗B C → ΩC/A → ΩC/B → 0is split exact, so when we apply HomMod(C)(_ ,M), the resulting sequence

0→ DerB(C,M)→ DerA(C,M)→ DerA(B,M)→ 0

is exact, which implies (2) in light of exactness of (3.7.3). Obviously (2) implies (3).For (3) implies (4) the point is that the extension in question clearly becomes thetrivial A-algebra extension when we precompose with f . For (4) implies (1), thepoint is that the algebra extension in (4) is the image of the identity map

Id ∈ HomMod(C)(ΩB/A ⊗B C,ΩB/A ⊗B C)

under the isomorphism

HomMod(C)(ΩB/A ⊗B C,ΩB/A ⊗B C) = DerA(B,ΩB/A ⊗B C)

followed by the map δ (for the sequence (3.7.3) with M = ΩB/A ⊗B C), so if itvanishes, exactness of (3.7.3) implies that Id lifts to an element

r ∈ HomMod(C)(ΩC/A,ΩB/A ⊗B C)

i.e. a retract of the left map of (3.7.1).

Corollary 3.7.3. Let f : A→ B, g : B → C be maps of rings (or sheaves of rings)with g surjective with kernel I. Then there is an exact sequence of C-modules

I/I2δ // ΩB/A ⊗B C // ΩC/A

// 0,(3.7.4)

where δ(i) := di⊗ 1. The following are equivalent:

(1) δ has a retract (i.e. δ is injective and (3.7.4) is split exact).(2) The rst innitesimal neighborhood B/I2 ∈ ExalA(C, I/I

2) is trivial.(3) The map g∗ : ExalA(C,M) → ExalA(B,M) is injective for every C-module

M .

Proof. For the rst statement: First note that g surjective implies ΩC/B = 0, henceDerB(C,M) = 0 for any C-module M (3.4). For any C-module M , the sequenceof C-modules obtained by applying HomMod(C)(_ ,M) to the sequence in questionis identied, via the isomorphism (3.5.2) of 3.5, and the isomorphism

ExalB(C,M) = HomMod(C)(I/I2,M)(3.7.5)

of Theorem 3.3.1 with the sequence

0→ DerA(C,M)→ DerA(B,M)→ ExalB(C,M),(3.7.6)

which is exact by Theorem 3.7.1, so the rst statement follows formally (Exercise 6).

For the next statement, note that Theorem 3.7.1 actually says that the exactsequence (3.7.6) is part of a long exact sequence

0→ DerA(C,M)→ DerA(B,M)→ ExalB(C,M)→ ExalA(C,M)→ ExalA(B,M).


If δ has a retract, then the exact sequence in the statement of the Theorem is actuallya split exact sequence

0 // I/I2δ // ΩB/A ⊗B C // ΩC/A

// 0

so when we apply HomMod(C)(_ ,M) the resulting sequence

0→ DerA(C,M)→ DerA(B,M)→ ExalB(C,M)→ 0

is already exact, which means in the long exact sequence above,

ExalB(C,M)→ ExalA(C,M)

is the zero map (equivalenlty,

ExalA(C,M)→ ExalA(B,M)

is injective). It is clear from this that (1) implies (2) and (3) (take M = I/I2).On the other hand, if we take M = I/I2 and note that Theorem 3.3.1 says that(3.7.5) takes B/I2 ∈ ExalB(C,M) to Id ∈ HomMod(C)(I/I

2, I/I2), then either (2)or (3) applied to the long exact sequence above with M = I/I2 ensures that Id ∈HomMod(C)(I/I

2, I/I2) has a lift to

HomMod(C)(ΩB/A ⊗B C, I/I2) = DerA(B,M)

(i.e. δ has a retract). Corollary 3.7.4. Let A be a ring, C a nite type (resp. nitely presented) A-algebra.Then ΩC/A is a nitely generated (resp. nitely presented) C-module.

Proof. By denition of nite type (resp. nitely presented), A → C factors asA → B → C where B = A[x1, . . . , xn] is a polynomial ring over A in nitely manyvariables and g : B → C is surjective (resp. surjective with kernel I nitely gen-erated, say by i1, . . . , im). The B-module ΩB/A is freely generated by dx1, . . . , dxn

(we will see this later in Lemma 6.2.1 though it is elementary), so ΩB/A ⊗B Cis the free C-module on generators dx1, . . . , dxn. By Corollary 3.7.3, the mapCn ∼= ΩB/A ⊗B C → ΩC/A taking dxi to d(g(xi)) ∈ ΩC/A is surjective, so ΩC/A

is nitely generated (resp. and the map Cm → ΩB/A ⊗B C taking the jth basisvector to dij ⊗ 1 surjects onto its kernel, so ΩC/A is nitely presented).

3.8. Geometric dierentials. For a map of ringed spaces f : X → Y , we setΩX/Y := ΩOX/f−1OY

. Using (3.2.1) we see that ΩX/Y,x = ΩOX,x/OY,f(x). Consider an

inverse limit system (fi : Xi → Yi)i of maps of ringed spaces with limit f : X → Yand projections pi : X → Xi, qi : Y → Yi. Then the natural map

lim−→

p∗iΩXi/Yi→ ΩX/Y(3.8.1)

is an isomorphism because the natural FlAn(X) morphism

lim−→

(p−1i f ♯i : f

−1q−1i OYi→ p−1i OXi

) → (f ♯ : f−1OY → OX)(3.8.2)

is an isomorphism by construction of inverse limits of ringed spaces. It is lessobvious, but still true, that (3.8.1) is an isomorphism for an inverse limit system


(fi : Xi → Y )i of maps of locally ringed spaces (with the inverse limit taken inmaps of locally ringed spaces2). In this case, (3.8.2) is not generally an isomorphism(for example because the sheaves of rings on the left might not have local stalks, incontrast to the sheaves of rings on the right). However, the maps

lim−→

p−1i OXi→ OX

lim−→

q−1i OYi→ OY

are localization morphisms (on stalks, each map is the localization at some primeideal) [G, Theorem 9], which is enough to concude that (3.8.1) is an isomorphism.

3.9. The case of schemes. Since these notes supposedly focus on the deformationtheory of schemes, it is worth making a few remarks about how the above generalconstructions with dierentials behave for schemes.

Theorem 3.9.1. Let f : A → B be a map of rings, X := SpecB, Y := SpecA,g : X → Y the map induced by f . Then the OX-module ΩX/Y (3.8) is naturallyisomorphic to the quasi-coherent OX-module Ω

∼B/A associated to the B-module ΩB/A.

Proof. From the denition of OY , we have a map of sheaves of rings A→ OY on Y ,where A is the constant sheaf associated to the ring A. The stalk of this map at apoint y ∈ Y (i.e. a prime ideal of A) is the localization A → Ay of A at y. Thereis a similar map B → OX of sheaves of rings on X, and a commutative diagram ofsheaves of rings

B // OX

A

f

OO

// g−1OY

g♯

OO

on X. In fact this diagram is a pushout because this can be checked on stalks andthe stalk of the above diagram at a point x ∈ X (i.e. a prime ideal of B) is thediagram

B // Bx

A

f

OO

// Ay

OO

(where y ∈ Y is the inverse image of the prime ideal x under f), which is easilyseen to be a pushout via the universal property of localization. Since dierentialscommute with direct limits" (Formula 3.6.3 above), we have

ΩX/Y := ΩOX/f−1OY

= ΩB/A ⊗B OX .

2This inverse limit always exists. If the fi are maps of schemes and the inverse limit system isnite (so the inverse limit exists in schemes), then the inverse limit in schemes and in locally ringedspaces is the same.


Since dierentials commute with inverse images" (the special case of Formula 3.2.2where f is the map from X to a point) we have

ΩB/A = ΩB/A.

But, as is well-known and clear from the construction of M∼, we have M∼ = M ⊗B

OX for any B-module M , so we nally get

ΩX/Y = ΩB/A ⊗B OX

= Ω∼B/A.

Corollary 3.9.2. Let f : X → Y be a map of schemes. Then ΩX/Y is a quasi-coherent OX-module. If f is of locally nite type (resp. presentation), then ΩX/Y islocally nitely generated (resp. presented).

Proof. The questions are local, so Theorem 3.9.1 immediately yields the rst state-ment and reduces the niteness statements to Corollary 3.7.4.

4. Commutative algebra

This section is a compendium of some results from commutative algebra used inother parts of the text. Proofs will be kept to a minimum as there are many goodreferences.

4.1. Artinian rings.

Denition 4. A ring A is called artinian i there is no innite strictly descendingchain of ideals of A.

Theorem 4.1.1. (1) Every artinian ring is noetherian.(2) Every artinian ring is a nite product of artinian local rings.(3) Every prime ideal in an artinian ring is maximal.(4) If (A,m) is an artinian local ring, then there is an n ∈ N such that mn = 0.

Proof. See [Eis, 2.14, 2.16]. For (4), note that, in any ring, the intersection of allprime ideals is the ideal of nilpotents, so (3) implies that every element of m isnilpotent, hence m is nilpotent because it is nitely generated in light of (1).

Denition 5. A small thickening is a surjection T ′ → T of artinian local rings withsquare-zero kernel I isomorphic to the residue eld of T as a T -module.

Lemma 4.1.2. Any surjection of artinian local rings with (non-zero) nilpotent ker-nel can be factored as a nite sequence of small thickenings.

Proof. Suppose f : T ′ → T is a surjection of artinian local rings with nilpotentkernel I. Since I is nilpotent, we must have I ⊆ m′, where m′ is the maximal ideal


of T ′. Note that m = f(m′) is the maximal ideal of T . By Theorem 4.1.1, there isan n ∈ N so that mn = 0. The quotient seqeuence

T ′ → T ′/mn−1I → · · · → T ′/mI → T = T ′/I

factors f into surjections of artinian local rings with square-zero kernels annihilatedby the maximal ideal of the codomain. By further factoring these surjections, wereduce to the case where mI = 0, so I is a (nite dimensional) vector space overthe residue eld K := T/m = T ′/m′. Any K-linear subspace of I is then an ideal ofA because it is an A-submodule of I since the A-module structure on I is given byrestriction of scalars along A→ K, so we can quotient by one-dimensional subspacesof I successively to obtain the desired factorization.

4.2. Regular sequences.

Denition 6. Let A be a ring, M an A-module. An element a ∈ A is called M -regular i scalar multiplication ·a : M → M is injective. A sequence a1, . . . , an ofelements of A is called an M -regular sequence i ai+1 is M/(a1, . . . , ai)M -regular fori = 0, . . . , n − 1. In particular, when M = A, an element a ∈ A is called regular ifit is not a zero divisor. A sequence (a1, . . . , an) of elements of A is called a regularsequence i ai+1 is regular in A/(a1, . . . , ai) for i = 0, . . . , n− 1.

Lemma 4.2.1. Let A → B be a surjection of rings whose kernel I is generatedby a regular sequence (a1, . . . , ar). Then the B-module I/I2 is freely generated bya1, . . . , ar.

Proof. Obviously I/I2 is generated as a B-module by the ai, so the issue is to provethat for any b1, . . . , br ∈ A with

∑i biai ∈ I2, it must be that b1, . . . , br ∈ I. Since

I2 is generated by aiaj : 1 ≤ i ≤ j ≤ n, we can write∑i

biai =∑i≤j

cijaiaj(4.2.1)

for some cij ∈ A. This implies

brar = crra2r

in the ring Ar := A/(a1, . . . , ar−1), which implies br = crrar in Ar (because ar is aregular element in Ar by denition of a regular sequence), so we can write

br = a1d1 + · · ·+ ar−1dr−1

in A for some di ∈ A (in particular br ∈ I). Plugging this into (4.2.1) and readingthe result in Ar−1 := A/(a1, . . . , ar−2), we nd

ar−1br−1 + ar−1ardr−1 = cr−1,r−1a2r−1 + cr−1,rar−1ar

in Ar−1. Again, we can cancel the regular element ar−1 ∈ Ar−1 from both sides andsolve for br−1 (in particular we nd br−1 ∈ I). After continuing down the regularsequence in this manner one eventually nds that every bi is in I.


4.3. Regular rings. Regular local rings (dened momentarily) are ubiquitous indeformation theory and the theory of smoothness and formal smoothness (5).

Theorem 4.3.1. Let (A,m, K) be a noetherian local ring of nite Krull dimensiond. Then the following are equivalent:

(1) dimK m/m2 = d.(2) m can be generated by d elements.(3) Ext>d

A (M,N) = 0 for all A-modules M,N .(4) There is an integer n such that Ext>n

A (M,N) = 0 for all A-modules M , N .(5) TorA>d(M,K) = 0 for all A-modules M .(6) TorAd+1(K,K) = 0.

(7) There exists an integer n such that TorA>n(A,K) = 0.(8) The natural map

Sym∗K m/m2 →∞⊕n=0

mn/mn+1

is an isomorphism of graded K-algebras.(9) m can be generated by a regular sequence.

Proof. The equivalence of (1) and (2) is Nakayama's Lemma. For the other equiv-alences, see [Mat, Theorem 41], [EGA, 0IV.17.1.1] or [Mat, Theorem 35]. The factthat (4) implies the other conditions is a famous theorem of Serre [Mat, 18.G, The-orem 45]. Denition 7. A noetherian local ring satisfying the equivalent conditions of theabove theorem is called a regular local ring. A ring A is called regular i Ap is aregular local ring for every prime p ∈ SpecA. Similarly, a scheme X is called regulari the local ring OX,x is regular for each x ∈ X.

Proposition 4.3.2. Let k be a eld. Then the polynomial ring k[x1, . . . , xn] isregular.

Proof. [Eis, 19.14], for example. Proposition 4.3.3. A noetherian local ring (A,m) is regular i its m-adic comple-

tion A := lim←−

A/mn is regular.

The main example of a complete regular local ring is the formal power series ringk[[x1, . . . , xn]] over a eld k.

Here are some standard facts about regular local rings:

Theorem 4.3.4. A regular local ring is...

(1) an integral domain.(2) normal.(3) Cohen Macaulay.


(4) a unique factorization domain.

Proof. [Mat, 17.F, Theorem 36]. The fact that a regular local ring is a u.f.d.is originally due to Auslander and Buchsbaum, but the easiest proof is due toI. Kaplanskysee [Mat, Theorem 48].

Proof. Since A is noetherian, A→ A is faithfully at and the maximal ideal of A ism = mA. It follows easily that the rst condition in Theorem 4.3.1 holds for A i itholds for A. Proposition 4.3.5. If A is a regular local ring, then Ap is a regular local ring forevery p ∈ SpecA. Consequently, a ring A is regular i Am is a regular local ring foreach maximal ideal m of A.

Proof. This follows easily from any of the homological" characterizations in Theo-rem 4.3.1 since A→ Ap is at. Lemma 4.3.6. Let (A,m) be a regular local ring. Suppose that a1, . . . , am ∈ m arelinearly independent in m/m2. Then a1, . . . , am is a regular sequence in A.

Proof. The hypothesis (plus Nakayama) implies that we can extend a1, . . . , am to aminimal set of generators a1, . . . , an ∈ m, n := dimA. The longer sequence a1, . . . , anis regular by [Mat, 17.F, Theorem 36]. Lemma 4.3.7. Let (A,m) be a regular local ring, I ⊆ m an ideal. The local ringA/I is regular i I can be generated by a regular sequence (of length equal to dimA−dimB).

Proof. [Mat, 17.F, Theorem 36] or [EGA, IV.19.1.2].

4.4. Flat, projective, and free modules. This section contains some standardfacts about at, projective, and free modules which we will use at various points.

Lemma 4.4.1. A nitely presented module (over an arbitrary ring) is at i it isprojective.

Proof. See the Corollaire in [B, X.5], for example. Lemma 4.4.2. Any projective module over a local ring is free.

Proof. In general this is a pretty tough theorem of Kaplansky [IK], but for a nitelygenerated module it is a pretty easy exercise with Nakayama's Lemma. Lemma 4.4.3. Let A be a ring, p a prime ideal of A, M a nitely presented A-module. Then the following are equivalent:

(1) There is f ∈ A \ p such that Mf is a free Af -module.(2) Mp is a free Ap-module.(3) Mp is a at Ap-module.


(4) Mp is a projective Ap-module.

Proof. The last three conditions are equivalent by Lemma 4.4.1 and Lemma 4.4.2because Mp is a nitely presented Ap-module. Obviously (1) implies (2), even with-out the niteness hypotheses, because extension of scalars takes free modules to freemodules. The key point for (2) =⇒ (1) is that nite presentation of an A-moduleM ensures that an A-module map from M to a ltered direct limit factors throughone of the structure maps to the limit (c.f. [B, X.5.8]). This ensures that an isomor-phism Mp

∼= Anp actually comes from an isomorphism Mf

∼= Anf at some stage of the

ltered limit over f ∈ A \ p. Lemma 4.4.4. Let A be a reduced ring, M a nitely generated A-module. Thefollowing are equivalent:

(1) M is locally free: for each p ∈ SpecA there is an f ∈ A \ p such that Mf isa free Af -module.

(2) M has locally constant rank: the rank function p 7→ dimk(p) M ⊗A k(p) is alocally constant function on SpecA.

If A is a Jacobson ring (i.e. a ring where every prime ideal is the intersection of themaximal ideals containing it), then these conditions are equivalent to

(3) The function m 7→ dimk(m)M ⊗A k(m) is a locally constant function on themaximal ideals of A (with the topology inherited from SpecA).

Remark 11. Every ring of nite type over Z or a eld is Jacobson. This statementis a variant of the Nullstellensatz [Eis, 4.5].

Proof. Clearly (1) implies (2) (even without assuming A reduced). Now supposeM has locally constant rank. Fix p ∈ SpecA and let us prove that M is free ona neighborhood of p. By passing to a neighborhood of p we can assume M hasconstant rank r. Since M is nitely generated, Nakayama's Lemma implies that wecan nd (after possibly passing to a smaller neighborhood of p) m1, . . . ,mr ∈ Msuch that the A-module map m : Ar → M sending ei to mi is surjective. LetK := Kerm. For any q ∈ SpecA, we then have an exact sequence

K ⊗A k(q)→ k(q)r →M ⊗A k(q)→ 0

of k(q) vector spaces. The surjection on the right must be an isomorphism becauseits domain and codomain have the same dimension (namely r by the constant rankr assumption), hence the left map must be the zero map, hence K ⊆ qAr. Sincethis is true for all q ∈ SpecA and A is reduced (hence the nilradical ∩qq consistssolely of 0) we have K = 0, hence m witnesses local freeness of M .

Now assume A is reduced Jacobson and M has locally constant rank at maximalideals. To prove M is locally free, it is enough to prove that it is locally free ona neighborhood of each maximal ideal because any prime ideal p is contained insome maximal ideal m and then any neighborhood of m is also a neighborhood ofp. But this can be argued exactly as above, replacing the arbitrary prime q with an


arbitrary maximal ideal n, then noting that the Jacobson hypothesis implies thatthe nilradical is the intersection of all maximal ideals.

4.5. Fiberwise criteria. Suppose f : X → Y is a map of schemes. It is oftenindispensible to know that various properties of f (or other data on X) can bechecked at each ber Xy of f . Results to this eect are called berwise criteria. Thecommutative algebra analog of this question (into which the geometric questionoften translates easily) is the following: Given a local map of local rings f : A→ B,we let B := B ⊗A k, where k is the residue eld of A. We often want to know thatsome property of f (or often some property of a B module M or map of B-modulesg : M → N) can be checked on B.

Most results along these lines are proved by noetherian approximation: One rstproves the result for f : A → B a map of noetherian local rings, then one uses theresult for noetherian rings to get a similar result for an arbitrary map f : A → Bof local rings, often at the cost of imposing various niteness conditions (essentiallynite presentation of f , for example). This is usually accomplished by writingA → B as a ltered direct limit of local maps Ai → Bi of local rings of essentiallynite type over Z (hence noetherian); one then applies the noetherian result toAi → Bi and gets it for A→ B by using the fact that most algebraic properties arepreserved under ltered direct limitsoften the only subtlety is to show that someproperty of A → B (or the additional data on B) can be arranged to hold for thenoetherian approximations" Ai → Bi.

The general noetherian approximation technique is explained very well in theCommutative Algebra part of the Stacks Projectsee, in particular, [SP, 121-122].Grothendieck gives a very general treatment in [EGA, IV.8]. Personally I muchprefer de Jong's expositionthe objection I have to the Grothendieck approach isthat he spends an incredible amount of time developing the theory of approximat-ing an arbitrary (though usually nitely presented) map of schemes f , but in theapplications he only really needs to know some basic stu about approximating alocal map of local rings of essentially nite presentation.

Lemma 4.5.1. Let A→ B be a local map of local rings of essentially nite presen-tation, g : M → N a map of nitely presented B-modules. Assume that N is atover A. Let k be the residue eld of A. Then the following are equivalent:

(1) g is injective and Cok g is at over A.(2) g ⊗A k : M ⊗A k → N ⊗A k is injective.

Proof. When A and B are noetherian, this is [Mat, 20.E] (no niteness hypothesis onf is needed). The general case is proved by noetherian approximationsee [EGA,IV.11.3.7] or [SP, Lemma 122.4]. Lemma 4.5.2. Let A → B be a local map of local rings of essentially nite pre-sentation, M a nitely presented B-module, b ∈ B. Let k be the residue eld of A.Assume that M is at over A. Let k be the residue eld of A. Assume that b⊗ 1 isM ⊗A k-regular. Then b is M-regular and M/bM is at over A.


Proof. Apply the previous lemma to ·b : M →M .

Lemma 4.5.3. Let A→ B be a local map of local rings of essentially nite presen-tation,

0→M ′ →M →M ′′ → 0(4.5.1)

an exact sequence of B-modules. Assume that M and M ′′ are nitely presented andM ′′ is at over A. Then M ′ is nitely presented.

Proof. When B is noetherian this is trivial. The general case is proved by noetherianapproximation [EGA, IV.11.3.9.1]one argues that there is a local map of local ringsA0 → B0, both of essentially nite type over Z (hence noetherian), an exact sequence

0→M ′0 →M0 →M ′′

0 → 0(4.5.2)

of nitely generated B0-modules with M ′′0 at over A0 (the fact that one can arrange

this atness is [SP, 122.3]), and a map A0 → A such that B = B0⊗A0 A and (4.5.1)equals (4.5.2) tensored over A0 with A (note that M ′′

0 is at over A so this tensoringpreserves the exactness and the nite presentation ofM ′ is immediate from the nitepresentation of M ′

0).

Lemma 4.5.4. Let A→ B be a local map of local rings of essentially nite presen-tation, M a nitely presented B-module, b1, . . . , bm ∈ B. Assume M is at over A.Let k be the residue eld of A. Assume that

b1 ⊗ 1, . . . , bm ⊗ 1

is an M ⊗A k-regular sequence. Then b1, . . . , bm is an M-regular sequence andM/(b1, . . . , bm)M is at over A.

Proof. Set Mi := M/(b1, . . . , bi)M . By Lemma 4.5.2, b1 is M -regular and M1 is atover A. We now want to apply the same lemma to conclude that b2 is M1-regularand M2 is at over A (and so on...). But we need to know that M1 is nitelypresented as a B-module. We at least know it is a quotient of the nitely presentedB-module M , so we can nd a surjection An →M1, then use Lemma 4.5.3and thefact that M1 is at over Ato conclude that the kernel of this surjection is nitelygenerated (even nitely presented), so we're good. Repeating this argument, wend that all the Mi are at over A and nitely presented and bi+1 is Mi-regular fori = 0, . . . , n− 1.

Lemma 4.5.5. Let A be a ring, u : M → N a map of A-modules with M nitelygenerated and N projective. For every x ∈ SpecA, the following are equivalent:

(1) ux : Mx → Nx has a retract.(2) u⊗A k(x) : M ⊗A k(x)→ N ⊗A k(x) is injective.(3) There is a sequence m1, . . . ,mn ∈ M whose images in Mx generate Mx and

a sequence f1, . . . , fn ∈ HomA(N,A) such that det(fi(u(mj))) /∈ x.(4) There is f ∈ A \ x such that uf : Mf → Nf has a retract.


Proof. This is a clever lemma of Grothendieck [EGA, IV0.19.1.2]. Obviously (1)implies (2), (4) implies (1), and (3) implies (2). To prove (2) implies (1) we canclearly replace u with ux and A with Ax to assume that A is local and x = m isits maximal ideal, k := A/m. We will use notation like u : M → N as shorthandfor u ⊗A k. Since the map u of k vector spaces is injective by assumption, it has aretract v : N →M . Since N is projective, we can lift in

N //

v''

Nv // M

M

OO

as indicated. Then w := vu is an endomorphism of M reducing to the identity modm, and to complete the proof it suces to show that w is an isomorphism (for thenr := w−1v retracts u). If the nitely generated module M were free, isomorphy forw would be obvious from isomorphy of w mod m (for then, after choosing a basisfor M , w would be given by a matrix with invertible determinant, hence would bean isomorphism by basic linear algebra), so we have proved the result assuming Mis free, and we can conclude the general case by arguing that M is free. Choose a(nitely generated) free A-module L and an A-module map f : L → M inducingan isomorphism of k vector spaces f : L → M . It now suces to prove that fis an isomorphism. First, f is surjective by Nakayama, so it suces to prove f isinjective. Set g := uf : L → N . Then g = uf is injective, so g has a retract bythe previously-handled free case." In particular g is injective and hence f must beinjective, so f is an isomorphism as desired.

For the other implications, rst write the projective module N as a summandof a free A-module F = N ⊕ P . Since M is nitely generated, u : M → N ⊆ Ffactors through some nitely generated free summand N ′ of F = N ′ ⊕ P ′. Thestatements we want to prove are clearly equivalent to the same statements withN replaced by F = N ⊕ P , but, for the same reason, they are equivalent to thesame statements with F = N ′ ⊕ P ′ replaced by N ′. We thus reduce to proving theremaining equivalences when N is a nitely generated free A-module.

For (1) implies (3), rst note that (1) implies the nitely generated module Mx

is a summand of the nitely generated free module Nx, hence Mx must also be freeand we can certainly nd m1, . . . ,mn ∈ M whose images in Mx are a basis andh1, . . . , hn ∈ HomAx(Nx, Ax) with dethiu(mj) not in xAx, by basic linear algebra.Since Nx is free and nitely generated, each of those hi can be written as fi/vi forvi ∈ A \ x. Replacing the hi with the fi only has the eect of multiplying thedeterminant by v1 · · · vn, which does not change the fact that this determinant isnot in x.

For (1) implies (4), let rx : Nx → Mx be a retract of ux. Since N is nitelygenerated and free, we have HomAx(Nx,Mx) = HomA(N,M)x, so we can writerx = w/h for some w : N → M , h ∈ A \ x. Since rxux = Id : Mx → Mx, we havewux = h : Mx → Mx, which means that for each m ∈ M , wu(m) − h ·m ∈ M is


annihilated by an element of A \x. Since M is nitely generated, we can hence nda single g ∈ A\x so that g(wu−h) : M →M is zero. If we set f := hg, then h−1wh

retracts uf : Mf → Nf .

We will use the following critère de platitude par bres :

Lemma 4.5.6. Let f : X → Y be a at morphism of schemes of locally nitepresentation, F a quasi-coherent sheaf on X of locally nite presentation, at overY . Then the following are equivalent:

(1) F is at as an OX-module.(2) F is locally free of locally nite rank.(3) For each point y ∈ Y , the restriction of F to the ber Xy := X×Y Spec k(y)

is locally free.

Proof. The rst two statements are equivalent by Lemma 4.4.3. For the equivalenceof the second and third conditions, apply [EGA, IV.11.3.10] with X = Y there equalto the X here, S there equal to X here, and g = h there equal to f here. The rstparts of a) and b) there hold trivially since we assume F is at over Y and f is at,so that theorem says that for x ∈ X, with image y := f(x) ∈ Y , we have F |Xy

is at at x ∈ Xy" i F is at at x (equivalently locally free in a neighborhood ofx)." Remark 12. We will only make use of Lemma 4.5.6 in the proof of Theorem 10.3.3.In particular, we will only use it in the case where f is a locally nite type mapof noetherian schemes. In this noetherian case, the result holds even without theniteness assumption on f and is a pretty easy consequence of the standard localcriteria of atness" from commutative algebra (c.f. [Mat, 20.C] or [Eis, 6.8]).

5. Formal smoothness

This section is a summary of Grothendieck's approach to smooth maps of schemes.We start by dening formally smooth maps of rings (and several variants) and es-tablish their basic properties (5.1). The formal nature of the denition triviallyensures that formally smooth ring maps enjoy some nice properties (closure undercomposition and pushout, for example). The denition is also suciently categoricalthat it makes sense in other contexts (for example, one can immediately say whatit means for a presheaf on the category of rings to be formally smooth). In fact,Grothendieck's original theory of formal smoothness is formulated in the languageof topological rings. In the interest of brevity and simplicity, we do not take thisapproach.

The rst nontrivial result is the statement that for a formally smooth ring mapA→ B, theB-module of Kähler dierentials ΩB/A (3) is projective (Corollary 5.3.3).This is basically [EGA, IV0.19.5.3]. Despite the fact that this seems to be one ofthe highlights of the theory, Grothendieck admits that the proof (which is buried ina multipage footnote) is encombrée de détails techniques." This is mostly due to


having to keep track of topological issuesthe proof given here is really not at alldicult.

Having dened a formally smooth map of rings, one can dene a smooth mapof rings to be a formally smooth map of rings which is of nite presentation. Animportantand rather nontrivialfact is that a smooth map of rings is at (it isnot true that a formally smooth map of rings is at). This is proved (using severalof the commutative algebra results from 4) in 5.10.

5.1. Denitions and rst properties.

Denition 8. A map of rings A → B is called formally smooth (resp. formallyétale, formally unramied) i there is a lift (resp. a unique lift, at most one lift) asindicated in any commutative square of rings

T Boo

~~T ′

OO

Aoo

OO(5.1.1)

where T ′ → T is surjective with square-zero kernel. A map of sheaves of ringsA → B is called formally smooth (resp. formally étale, formally unramied) ithere is locally a lift (resp. there is globally a unique lift, there is at most onelift even locally) as indicated in every diagram of sheaves of rings as above whereT ′ → T is surjective with square-zero kernel. A map of sheaves of rings A → Bis called stalkwise formally smooth (resp. stalkwise formally étale) i the map onstalks Ax → Bx is formally smooth (resp. formally étale) for each point x.

One can show (Exericse 7) that formally smooth (resp. étale)" implies stalkwiseformally smooth (resp. étale)." One can replace square-zero kernel" with nilpotentkernel" in the above denition without changing the meanings of the terms becauseany surjection with nilpotent kernel can be factored as a nite sequence of surjectionswith square zero kernels and one can make the lifts one step at a time. Clearlyformally étale" is the same as formally smooth and formally unramied".

Proposition 5.1.1. (1) Formally smooth, formally étale, and formally unram-ied maps (of rings or sheaves of rings) are closed under composition.

(2) Formally smooth, formally étale, and formally unramied maps (of rings orsheaves of rings) are closed under pushout.

(3) An epimorphism of rings (or sheaves of rings) is formally unramied.(4) If f : A→ B is formally unramied and g : B → C has gf formally smooth

(resp. formally étale), then g is formally smooth (resp. formally étale).(5) If f : A → B is formally étale, then a map of rings (or sheaves of rings)

g : B → C is formally smooth (resp. formally étale) i gf is formally smooth(resp. formally étale).

(6) If (Bi) is any direct limit system of formally étale A-algebras (A a ring orsheaf of rings), then the direct limit B is a formally étale A-algebra.


(7) If Bi is a set of formally smooth A-algebras, the tensor product (coproductin the category of A-algebras) B := ⊗ABi is a formally smooth A-algebra.(For sheaves of rings, one should assume that Bi is a nite set.)

(8) A transnite composition of formally smooth ring maps is formally smooth.(The meaning of trannite composition" will be explained in the proof. Thereis no analog for sheaves of rings.)

Proof. Statements (1) and (2) are trivial exercises with the denition. Statement(3) is obvious, since even the lower right triangle in a diagram (5.1.1) can have atmost one completion when A→ B is an epimorphism.

For (4), let us prove that g is formally smooth (resp. formally étale) under theassumptions on f and gf . Fix a square-zero surjection T ′ → T and consider threesolid diagrams as below:

T Coo

yyT ′

OO

Bh

oo

g

OO T Coo

k

yyT ′

OO

Ahf

oo

gf

OO T Boo

yyT ′

OO

Ahf

oo

f

OO

Since gf is formally smooth (resp. formally étale), the middle diagram has a com-pletion k as indicated (at least locally in the sheaves of rings case) (resp. a uniquecompletion). We then claim that k also completes the left diagramthe issue is toshow that h = kg. But both h and kg will complete the right diagram, so h = kgbecause f is formally unramied. Statement (5) is immediate from (4) and (1).

For (6), it suces to treat coproducts and coequalizers. The universal propertyof coproducts says that the data of a solid diagram as on the left below (i.e. anA-algebra map from B to T ) is the same thing as a set of solid diagrams as on theright below (one for each i) and a completion on the left is the same thing as acompletion of each diagram on the right, so the result is trivial general nonsense.

T Boo

~~T ′

OO

Aoo

OO T Bioo

~~T ′

OO

Aoo

OO

The same argument proves (7), but the issue in the sheaves case is that one has tond a small enough neighborhood of any given point where all the right diagramscan be completed, which we can certainly do if there are only nitely many diagramsto complete. We still need to prove that a coequalizer of formally étale A-algebrasis formally étale, so consider a formally étale A-algebra map f : A → B and twoA-algebra maps g1, g2 : B ⇒ C so that the common map g := g1f = g2f is formallyétale. Let h : C → D be the coequalizer. We need to show that hg is formallyétale. By the universal property of coequalizers, a solid diagram of rings as on the


left below

T Doo

C

h

OO

B

OOg1, g2

OO

T ′

OO

Aoo

f

OO

T Caoo

l

B

OOg1, g2

OO

T ′

OO

Aoo

f

OO

T Bkoo

~~T ′

OO

Aoo

f

OO

is the same thing as a solid diagram of rings in the middle where ag1 = ag2 (call thiscommon map k) and a completion as indictaed in the left diagram is the same thingas a completion as indicated in the middle diagram with lg1 = lg2. When T ′ → T isa square-zero surjection we can certainly complete the middle diagram as indicatedbecause g is formally étale. The condition lg1 = lg2 holds because both lg1 and lg2will complete the right diagram and f is formally étale.

For (8), suppose I is a well-ordered set (ordinal) and

Bi : i ∈ I, bij : Bi → Bj : i, j ∈ I, i < jis a direct-limit-preserving functor from I to rings such that each bij is formallysmooth. Let 0 ∈ I denote the minimum element and let B denote the direct limitof this functor. We claim that A := B0 → B is formally smooth. A solid diagramas on the left below

T Boo

~~T ′

OO

Aoo

OO T Biaioo

li

~~T ′

OO

Aoo

OO

is the same thing as a set of solid diagrams as on the right (one for each i ∈ I) suchthat ai = ajbij for every i, j ∈ I with i < j (and so that a0 : A = B0 → T is thecomposition A→ T in the left diagram). A completion on the left is the same thingas a completion li of each diagram on the right such that li = ljbij for all i, j ∈ Iwith i < j. If there were no set li : i ∈ I of such li, then since I is well-ordered,there would be some smallest element k ∈ I such that we have completions li ason the right for all i < k satisfying li = ljbij for all i < j < k, but we have nocompletion lk satisfying li = lkbik for all i < k. This cannot happen if k is a limitpoint in I because Bk is the direct limit of the Bi with i < k and we could just takelk to be the direct limit of the li over all i < k, nor can this happen if k = m+1 is asuccessor because Bm → Bk is formally smooth, so we can complete in the diagram

T Bkoo

lk

~~T ′

OO

Bmlmoo

bmk

OO

as indicated.


Remark 13. The above proposition is just a formal statement about a class ofmaps in a category which is dened by having the (local) right lifting property"with respect to another class of maps. This sort of denition occurs frequentlyin Quillen's theory of model categories, for example. Conspicuously missing fromthe above proposition is the statement a ltered direct limit of formally smooth A-algebras is formally smooth." Indeed, it is not true that a general ltered direct limitof formally smooth algebras is formally smooth (see Example 2 in 6.2). However,ltered direct limits of formally smooth A-algebras are pretty nice" A-algebras andwe will have occassion to make use of them frequently.

Lemma 5.1.2. A map of rings (resp. sheaves of rings) A→ B is formally smoothi every A-algebra extension of B (3.3) by any B-module is trivial (resp. locallytrivial).

Proof. Given an A-algebra extension B′ → B, formal smoothness implies that thereis a lift (at least locally in the sheaf case) in the diagram

B B

~~B′

OO

Aoo

OO

which is the same thing as an A-algebra section of B′ → B (a trivialization of thealgebra extension). Conversely, given a solid diagram of rings (or sheaves of rings)

T Boo

~~T ′

OO

Aoo

OO(5.1.2)

where T ′ → T is surjective with square-zero kernel I, we obtain an A-algebra exten-sion of B by I (regarding I as a B-module via restriction of scalars along B → Tand the usual T -module structure on I) by pulling back the sequence

0→ I → T ′ → T → 0

along B → T . That is, if we set B′ := T ′×T B (this really means the bered productof rings or sheaves of rings, not the tensor product), we obtain a commutativediagram

0 // I // T ′ // T // 0

0 // I // B′

π1

OO

π2 //// B

OO

// 0

A

OO`ÀAAAAAAA

where the rows are A-algebra extensions. A lift (resp. local lift) in (5.1.2) is thesame thing as a trivialization (resp. local trivialization) of the algebra extension onthe bottom row.


Lemma 5.1.3. Consider a commutative diagram of rings (or sheaves of rings)

T Boo

l~~T ′

OO

Aoo

OO

where T ′ → T is surjective with square-zero kernel J . Suppose that there is a lift las indicated. Then the map taking an A-linear derivation D : B → J to the functionl + D : B → T ′ establishes a bijection between DerA(B, J) and the set of all liftsas indicated. (Here J is regarded as a B-module using restriction of scalars alongB → T and the T -module structure on J discussed in 2.2.)

Proof. We rst prove that if s, t are two lifts as indicated their dierence D := s− t :B → J is an A-linear derivation. The point is mostly that the way we are regardingJ as a B-module ensures b · j = s(b)j = t(b)j for j a local section J , b a local sectionof B. Clearly D kills anything in the image of A, so the issue is to check the LeibnitzRule, which is an easy computation:

D(b1b2) = s(b1)s(b2)− t(b1)t(b2)

= s(b1)(s(b2)− t(b2)) + t(b2)(s(b1)− t(b1))

= b1 ·D(b2) + b2 ·D(b1).

It remains to prove that for any A-linear derivation D : B → J and any lift l, themap l + D : B → T ′ is also a lift. Clearly it makes the diagram commute as adiagram of sets (sheaves of sets) because D kills A and maps B into the kernel ofT ′ → T , so the only issue is to check that l +D is actually a ring homomorphism.We compute

(l +D)(b1b2) = l(b1b2) +D(b1b2)

= l(b1)l(b2) + b1 ·D(b2) + b2 ·D(b1)

= l(b1)l(b2) + l(b1)D(b2) + l(b2)Db1 + (Db1)(Db2)

= (l(b1) +D(b1))(l(b2) +D(b2))

= (l +D)(b1)(l +D)(b2),

using the Leibnitz Rule for D, the denition of the B-module structure on J , andthe fact that D(b1)D(b2) = 0 because D takes values in J and J2 = 0.

Corollary 5.1.4. A map of rings (or sheaves of rings) A→ B is formally unramiedi ΩB/A = 0, so A→ B is formally étale i it is formally smooth and ΩB/A = 0.

Proof. If ΩB/A = 0, then DerA(B,M) = HomB(ΩB/A,M) = 0 for every B-moduleM , so by the lemma there can be at most one lift in any diagram (5.1.1), so A→ Bif formally unramied. Now suppose A → B is formally unramied. Let M be a


B-module, B[M ] the trivial square zero extension of B by M . The diagram

B B

l||B[M ]

OO

Aoo

OO

has a lift given by b 7→ (b, 0), so the lemma yields a bijection between DerA(B,M) =HomB(ΩB/A,M) and the set of all lifts in this diagram. Since A → B is formallyunramied there is at most one such lift, hence there is only one map from ΩB/A toany B-module M (the zero map), hence ΩB/A = 0.

5.2. Examples. Here we collect the main examples of formally smooth ring maps.

Denition 9. An A-algebra B is called free i there is a subset S ⊆ B so thatthe natural map A[S] → B is an isomorphism of A-algebras. Here A[S] is thepolynomial ring over A in variables xs : s ∈ S and the natural map sends xs tos ∈ B.

Proposition 5.2.1. A free A-algebra is formally smooth.

Proof. Following the notation of the denition, one can nd a lift in a diagram(5.1.1) by choosing any lifts of the images of elements of S in T to T ′.

Proposition 5.2.2. Let A be a ring (or sheaf of rings), S ⊆ A a subset (subsheaf).Then the localization A→ S−1A of A at S is formally étale.

Proof. The universal property of A→ S−1A is as follows: to give a map of sheavesof rings S−1A → B is to give a map of sheaves of rings A → B taking each localsection of S to a unit in B. In particular, A → S−1A is an epimorphism in thecategory of sheaves of rings. Therefore, a solid diagram of rings

T S−1Aoo

||T ′

OO

Aoo

OO

is nothing but a pair of maps A → T ′, T ′ → T so that the composition A → Ttakes local sections of S to units of T and such a diagram has a (necessarily unique)completion as indicated i A → T ′ already takes local sections of S to units of T ′.If T ′ → T is a surjection with nilpotent kernel, then this must be the case becausewe saw in Proposition 2.1.2 that any local section of T ′ mapping to a unit in T isalready a unit in T ′.

Corollary 5.2.3. Suppose f : A → B is a formally smooth (resp. formally étale)ring map. For any x ∈ SpecB, if we set y := f−1(x) ∈ SpecA, then fx : Ay → Bx

is formally smooth (resp. formally étale).


Proof. We have a commutative diagram of rings

Af //

i

B

j

Ayfx

// Bx

where the localizations i, j are formally étale by Proposition 5.2.2. The map jf :A → Bx is hence formally smooth (resp. formaly étale) by Proposition 5.1.1(1),hence fx is formally smooth (resp. formally étale) by Proposition 5.1.1(5) for thecomposition fxi.

A more pathological" example of formally étale ring maps is given in Exercise 8.

5.3. Formal smoothness and dierentials. Formal smoothness has many con-sequences for the sheaf of Kähler dierentials which we now summarize. Recall thatan object P of an abelian category A is called projective i any A-diagram

P

~~ M // N

with M → N an epimorphism can be completed as indicated. If A is the abeliancategory of modules over a sheaf of rings on a topological space, then we say thatP is locally projective i any such diagram can be completed locally as indicated.

Lemma 5.3.1. Let X = SpecA be an ane scheme, P an A-module. If the quasi-coherent OX-module P

∼ is locally projective in the above sense, then P is a projectiveA-module.

Proof. Suppose

P

~~ M // N

is a diagram of A-modules with M → N surjective. Let K be the kernel of M → N .To lift as indicated in this diagram is the same thing as lifting in the correspondingdiagram

P∼

|| M∼ // N∼

of OX-modules. The map M∼ → N∼ is still surjective and its kernel is K∼, so if P∼

is locally projective we can nd a lift as indicated, at least locally. The dierence of


two lifts is an OX-module map P∼ → K∼, so the obstruction to patching togetherlocal lifts into a global lift lies in

H1(X,H om(P∼, K∼)) = H1(X,Hom(P,K)∼),

which vanishes since X is ane and Hom(P,K)∼ is quasi-coherent. Theorem 5.3.2. Let f : A → B be a formally smooth ring map, g : B → C asurjective map of rings with kernel I such that gf : A→ C is also formally smooth.Then I/I2 is a projective C-module. (For sheaves of rings, one concludes only thatI/I2 is locally projective.)

Proof. Fix a diagram of C-modules

I/I2

l

M // N

with M → N surjective. We must construct a lift l as indicated. By pushing outthe rst innitesimal neighborhood, we obtain a diagram of C-modules with exactrows

0 // I/I2

// B/I2 //

C // 0

0 // N // E // C // 0

where the square on the right is a commutative diagram of B-algebras. Let h : B →E be the B-algebra structure map (the composition of B → B/I2 and B/I2 → E).Note that h takes I ⊆ B into N ⊆ E. Since gf is formally smooth, there is an A-algebra section s : C → E of E → C (c.f. Lemma 5.1.2) and hence an isomorphismE ∼= C[N ] of rings over C and under A so that the inclusion N → E is identiedwith the natural map N → C[N ]. Regard h as a ring map B → C[N ] via thisisomorphism.

The surjection M → N yields a surjection C[M ]→ C[N ] of rings whose kernel iscontained in M ⊆ C[M ] and hence is certainly square zero, so, since f is formallysmooth, there is a lift m as indicated in the diagram:

C[N ] Bhoo

m

C[M ]

OO

Aoo

f

OO

Since h takes I ⊆ B into N ⊆ C[N ], m must take I ⊆ B into M ⊆ C[M ], hence mmust kill I2 and m descends to a map l : I/I2 →M providing the desired lift. Corollary 5.3.3. Suppose A → C is a formally smooth map of rings. Then ΩC/A

is a projective C-module. (For sheaves of rings, one concludes only that ΩC/A islocally projective.)


Proof. Since A→ C is formally smooth and formally smooth maps are closed undercomposition and base change (Proposition 5.1.1), the map A→ C ⊗A C is formallysmooth. The multiplication map m : C ⊗A C → C is surjective and we have I/I2 =ΩC/A where I := Kerm (Proposition 3.1.1). The composition A→ C ⊗A C → C isthe original ring map A→ C, so it is formally smooth, so the theorem says ΩC/A isprojective.

Proposition 5.3.4. Let A→ B be a ring map, B → C a surjective ring map withkernel I. Assume A→ C is formally smooth. Then the sequence of C-modules

0 // I/I2δ // ΩB/A ⊗B C // ΩC/A

// 0

is split exact.

Variants:

(1) If one only assumes C is a ltered direct limit of formally smooth A-algebras,then one concludes only that the sequence is exact and ΩC/A is a at C-module.

(2) For sheaves of rings, one concludes only that the sequence is exact and locallysplit.

(3) If one only assumes A → C is stalkwise formally smooth, then one onlyconcludes that the sequence is exact and splits on stalks.

(4) If one only assumes each stalk of A→ C is a ltered direct limit of formallysmooth Ax-algebras, then one concludes only that the sequence is exact andΩC/A is a at C-module.

Proof. The sequence is split exact by Corollary 3.7.3 (using (2) implies (1)) be-cause A → C formally smooth implies ExalA(C,M) = 0 for every C-module M(Lemma 5.1.2). The variant for sheaves is proved identically. If A→ C is a ltereddirect limit of formally smooth A-algebras Ci, then if we set Bi := B ×C Ci, thenwe have commutative diagrams of rings

A // Bi

// Ci

A // B // C

such that the bottom row is the ltered direct limit of the top rows. Furthermore,each Bi → Ci is surjective and I is the ltered direct limit of the Bi-modulesIi := Ker(Bi → Ci). Similarly, formation of dierentials commutes with ltereddirect limits, so the sequence of interest to us is the ltered direct limit of thesequences

0 // Ii/I2i

δ // ΩBi/A ⊗BiCi

// ΩCi/A// 0

associated to the top rows, each of which is split exact by the original proposition.Filtered direct limits are exact, so in particular this means our sequence is exact.Furthermore, ΩC/A is the ltered direct limit of the ΩCi/A and each ΩCi/A is projective


(hence at) by Corollary 5.3.3, hence ΩC/A is at (a ltered direct limit of atmodules is at because tensor products commute with ltered direct limits).

The other variants are proved as follows: Exactness can be checked at stalks,where the sequence becomes

0 // Ix/I2x

δ // ΩBx/Ax ⊗Bx Cx// ΩCx/Ax

// 0

and Ix is the kernel of the surjection Bx → Cx. If Ax → Cx is formally smooth, theabove is split exact by the original version of the proposition, and if Cx is a ltereddirect limit of formally smooth Ax-algebras, then the above is a ltered direct limitof split exact sequences by the variant discussed above, so is, in particular, exact.Flatness of ΩC/A can be checked on stalks similarly.

Remark 14. Proposition 5.3.4 can be used to give another proof of Corollary 5.3.3which does not use Proposition 3.1.1. We can factor any ring map A → C asA→ B → C where B is a free A-algebra (polynomial ring), for example by takingB = A[C]. If A→ C is formally smooth, then

0 // I/I2δ // ΩB/A ⊗B C // ΩC/A

// 0

is split exact by Proposition 5.3.4, so ΩC/A is a direct summand of ΩB/A⊗BC, henceit is projective because ΩB/A is a free B-module when B is a free A-algebra (c.f.Lemma 6.2.1 below).

Proposition 5.3.5. Let A→ B → C be a ring maps. Assume B → C is formallysmooth. Then the sequence of C-modules

0 // ΩB/A ⊗B C // ΩC/A// ΩC/B

// 0

is split exact.

Variants: Analogous to those of Proposition 5.3.4.

Proof. By Corollary 3.7.2 it suces to prove that the natural map

ExalB(C,M)→ ExalA(C,M)

is injective for each C-module M , but this holds trivially because formal smoothnessof B → C implies ExalB(C,M) is zero for every M (Lemma 5.1.2).

5.4. The formal Jacobian criterion. There is a kind of converse to Proposi-tion 5.3.4the formal Jacobian criterionwhich is very useful in practice for de-ciding when a given ring map A→ C is formally smooth:

Theorem 5.4.1. Let f : A→ B be a formally smooth map of rings and let g : B →C be a surjective map of rings with kernel I. Then the following are equivalent:

(1) gf : A→ C is formally smooth.


(2) The sequence

0→ I/I2 → ΩB/A ⊗B C → ΩC/A → 0

is exact (by Corollary 3.7.3 the only issue is injectivity of the left map) andΩC/A is projective.

(3) The sequence

0→ I/I2 → ΩB/A ⊗B C → ΩC/A → 0

is split exact.

Proof. (1) implies (2) by Proposition 5.3.4 and Corollary 5.3.3 and (2) clearly implies(3). For (3) implies (1), rst note that, by the universal property of the quotientmap g : B → C, a solid commutative diagram of rings as on the left below is thesame thing as a solid commutative diagram of rings as on the right below

T Coo

~~T ′

OO

Aoo

gf

OO T Bhoo

l

~~T ′

OO

Aoo

f

OO

where I ⊆ Kerh. Similarly, a lift as indicated in the left diagram is the same thingas a lift as indicated in the right diagram such that I ⊆ Ker l. Now suppose T ′ → Tis surjective with square zero kernel J . Note that J has a T -module structure andhence a C-module structure by restriction of scalars along C → T . Then since f isformally smooth, we can certainly nd some lift l in the right diagram (pick one),but the issue is to arrange that l kills I. Since h kills I and the diagram commutes,we know that l takes I into J ⊆ T ′ hence it kills I2 and we have a C-linear mapl : I/I2 → J . The split exactness hypothesis (3) ensures that our sequence staysexact applying Hom(_ , J), so we can nd a lift as indicated in the diagram ofC-modules:

ΩB/A ⊗B C

$$I/I2

OO

l // J

As in 3.5, this lift can be viewed as an A-linear derivation D : B → J (any suchderivation must kill I2) and the fact that the diagram commutes says exactly thatD|I/I2 : I/I2 → J is our original map l. As in Lemma 5.1.3, we can replace the liftl : B → T ′ with the lift l − D : B → T ′ to obtain a new lift in the right diagramwhich now kills I, hence furnishes a lift in the left diagram.

5.5. The case of schemes. The treatment of formal smoothness for schemes isslightly dierent from the way we set up formal smoothness for maps of sheaves ofrings on a xed space.


Denition 10. A map of schemes f : X → Y if called formally smooth (resp.formally étale, formally unramied) i there is a lift (resp. a unique lift, at most onelift) in any solid diagram of schemes

T //

X

f

T ′

>>

// Y

where T → T ′ is a square zero closed embedding of ane schemes.

Remark 15. It is clear from the denitions that a map of rings A→ B is formallysmooth (resp. formally étale, formally unramied) in the sense of Denition 8 ithe corresponding map of ane schemes SpecB → SpecA is formally smooth (resp.formally étale, formally unramied) in the sense of Denition 10 above.

Denition 11. A map of rings A → B is called smooth (resp. étale, unramied)i it is formally smooth (resp. formally étale, formally unramied) and of nitepresentation.

Denition 12. A map of schemes f : X → Y is called smooth (resp. étale, un-ramied) i it is formally smooth (resp. formally étale, formally unramied) and oflocally nite presentation.

Remark 16. The results analogous to those of Proposition 5.1.1 carry over to thecase of schemes by the same formal arguments: For example, formally smooth (hencealso smooth) maps are stable under base change, a composition of formally smoothmaps is formally smooth, etc. Similar results also hold for smooth maps becausemaps of schemes of locally nite presentation are closed under composition and basechange.

Example 1. An open embedding f : U → Y is étale. Clearly such a map islocally of nite presentation. Next note that a map T ′ → Y factors (necessarilyuniquely) through f i this is true on the level of underlying topological spaces.Formal étaleness is clear from this latter fact together with the fact that a squarezero thickening of schemes is an isomorphism on topological spaces (Lemma 2.1.1).

Formal smoothness (and also smoothness) is local on the map:

Lemma 5.5.1. Consider a commutative diagram of schemes

U //

g

X

f

V // Y

where the horizontal arrows are Zariski open covers. Then f is formally smooth(resp. smooth) i g is formally smooth (resp. smooth).

Proof. It is clear from the denition of locally nitely presented" that f is locallynitely presented i g is locally nitely presented, so we need only address the issue


of formal smoothness. If f is formally smooth, then its base change f−1(V ) → Vis formally smooth, hence g is formally smooth because U → f−1(V ) is a disjointunion of open embeddings (a renement of the cover f−1(V ) of X). Now supposeg is formally smooth and we want to show that f is formally smooth. Consider acommutative diagram

Ti //

X

f

T ′

>>

// Y

where T → T ′ is a square zero closed embedding of ane schemes with ideal J . LetW be the open cover of T obtained by pulling back U . After passing to a ner coverif necessary, we can assume W is a cover by ane opens. We can also think of W asan open cover of T ′ (on the level of spaces) since T and T ′ have the same topologicalspace (Lemma 2.1.1); let us denote this cover of T ′ by W ′ when we think of it as acover of T ′ by open subschemes. The mapW → W ′ is a disjoint union of square-zerothickenings of ane schemes (here one uses Proposition 2.2.2 to know that a squarezero thickening of an ane scheme is ane) and we have a commutative diagram

W //

U //

g

X

f

W ′

>>

// V // Y

so we can lift as indicated because g is formally smooth. This means that we can liftin the original diagram, at least locally on T ′. But any two lifts dier by a section ofH om(i∗ΩX/Y , J) (Lemma 5.1.3) so the only obstruction to choosing our local liftsin such a way that they will patch lies in

H1(T,H om(i∗ΩX/Y , J)),

which vanishes because T is ane and H om(i∗ΩX/Y , J) is quasi-coherent.

Lemma 5.5.2. If f : X → Y is a formally smooth (resp. formally étale) map ofschemes, then fx : OY,f(x) → OX,x is a formally smooth (resp. formally étale) mapof rings for every x ∈ X.

Proof. Suppose f is formally smooth (resp. formally étale). Fix x ∈ X. We cannd an ane open neighborhood V = SpecA of f(x) in Y and an ane openneighborhood U = SpecB of x in f−1(V ) ⊆ X. The map f |U : SpecB → SpecA isalso formally smooth (resp. formally étale) because the open inclusions U ⊆ X andV ⊆ Y are (formally) étale (c.f. Example 1, Proposition 5.1.1(5)). The correspondingmap of rings A → B is hence formally smooth (resp. formally étale) (Remark 15),so we reduce to Corollary 5.2.3.

Proposition 5.5.3. Let f : X → Y be a formally smooth (resp. smooth) map ofschemes. Then X has a cover by open anes Ui = SpecBi such that ΩX/Y |Ui = M∼

i


for Mi a projective Bi-module3 (resp. a free Bi-module of nite rank). If f is formally

étale, then ΩX/Y = 0.

Proof. We can work locally near some point x ∈ X. Let V = SpecA be an aneopen neighborhood of f(x) in Y and let U = SpecB be an ane open neighborhoodof x in f−1(V ). The map f |U : U → V is formally smooth (resp. smooth) becauseit is a composition of the open embedding U → f−1(V ) and the map f |f−1(V ) :f−1(V ) → V , which is formally smooth (resp. smooth) since it is a base changeof f . It is clear from the denitions that formal smoothness of f |U implies formalsmoothness of the corresponding ring map A → B and that we can take A → Bnitely presented when f is of locally nite presentation, hence A → B can andwill be taken smooth when f is smooth. We then have ΩX/Y |U = ΩU/V = Ω∼B/A byTheorem 3.9.1 and ΩB/A is a projective B-module by Corollary 5.3.3 (resp. and ΩB/A

is of locally nite presentation (Corollary 3.7.4), so it is locally free of nite rankafter possibly shrinking B to a smaller ane open neighborhood (Lemma 4.4.3).The statement about formally étale maps is proved the same way by arguing thatf |U is formally étale, hence A→ B is formally étale, hence ΩB/A =, hence ΩX/Y = 0because this can be checked locally.

5.6. The smooth locus is open. In this section we will prove that the smooth lo-cus of a locally nitely presented map of schemes is open. We start by combining theformal Jacobian criterion (Theorem 5.4.1) and the berwise criterion for retraction(Lemma 4.5.5):

Theorem 5.6.1. Let f : A → B be a formally smooth map of rings and let g :B → C be a surjective map of rings with kernel I. Assume that I/I2 is a nitelygenerated C-module. Consider the usual sequence of C-modules

0→ I/I2 → ΩB/C ⊗B C → ΩC/A → 0(5.6.1)

(which may not be exact on the left). For x ∈ SpecC with images y ∈ SpecB,z ∈ SpecA, the following are equivalent:

(1) Az → Cx is formally smooth.(2) The stalk of (5.6.1) at x is split exact.(3) The ber of (5.6.1) at x is exacti.e. I/I2 ⊗C k(x) → ΩB/A ⊗B k(x) is

injective.(4) The sequence (5.6.1) is split exact on a neighborhood of x in SpecC.(5) There is a neighborhood of x in SpecC formally smooth over Ai.e. an

element c ∈ C \ x such that A→ Cc is formally smooth.

The set of x ∈ SpecC satisfying these equivalent conditions is open in SpecC.

Proof. Since f is formally smooth, ΩB/A is a projective B-module (Corollary 5.3.3),so ΩB/A⊗BC is a projective C-module. Since we assume I/I2 is a nitely generated

3In other words, ΩX/Y is Qco locally projective in the sense of Denition 27.


C-module, we can apply Lemma 4.5.5 to the map I/I2 → ΩB/A ⊗B C to obtain theequivalence of (2), (3), and (4). Since the stalk of (5.6.1) at x is the usual sequence

0→ Ix/I2x → ΩBy/Az ⊗By Cx → ΩCx/Az → 0

associated to the map Az → By (which is formally smooth by Corollary 5.2.3 sincef is formally smooth) and the surjection By → Cx, (1) is equivalent to (2) byTheorem 5.4.1. The same theorem yields the equivalence of (4) and (5). It is clearthat (4) and (5) are open conditions. Corollary 5.6.2. Let A → C be a nitely presented ring homomorphism. Forx ∈ SpecC with image z ∈ SpecA, the following are equivalent:

(1) Az → Cx is formally smooth.(2) There is an f ∈ C \ x such that A→ Cf is smooth.

Proof. Apply the theorem to any A-algebra surjection B := A[x1, . . . , xn]→ C withnitely generated kernel I. Corollary 5.6.3. Let f : X → Y be a map of schemes of locally nite presentation.For x ∈ X, the following are equivalent:

(1) fx : OY,f(x) → OX,x is formally smooth.(2) There is a neighborhood U of x in X such that f |U : U → Y is smooth.

The set of x ∈ X satisfying these equivalent conditions is open.

Proof. The question is local so it reduces immediately to the previous corollary.

5.7. Formal smoothness and eld extensions. In this section we discuss formalsmoothness for eld extensions.

Denition 13. Let k be a eld. A k-algebra A is called separable i A ⊗k K is areduced ring (has no non-zero nilpotents) for each eld extension k → K.

Remark 17. Any eld extension k → K is a ltered direct limit of nitely generated(though perhaps non-algebraic) eld extensions k → L. Tensor products commutewith ltered direct limits. If A is a ltered direct limit of rings Ai and A has anontrivial nilpotent, then clearly one of the Ai must have a nontrivial nilpotent.Evidently then, we can replace each eld extension" with each nitely generatedeld extension" in the above denition without changing the meaning of separable.

Lemma 5.7.1. Let k → K be a nite eld extension. Then the following areequivalent:

(1) k → K is unramied (i.e. formally unramied)(2) ΩK/k = 0(3) k → K is étale (i.e. formally étale).(4) k → K is smooth (i.e. formally smooth).(5) k → K is separable.


Proof. (1) and (2) are always equivalent (Corollary 5.1.4), so we will forget about(1) and just work with (2).

Let us rst treat the case where there is a primitive element : a single elementζ ∈ K generating K over k. Then we obtain a presentation k[x] → K of K as ak-algebra by mapping x to ζ. The kernel of k[x]→ K is the ideal generated by theminimal polynomial f(x) ∈ k[x] for ζ over k. Since k[x] is (formally) smooth over kand any K module is projective, Theorem 5.4.1 says we have an exact sequence

(f)/(f 2)δ // Ωk[x]/k ⊗k[x] K // ΩK/k(5.7.1)

of K vector spaces and k → K is (formally) smooth i δ is injective. Note that(f)/(f 2) = (f)⊗k[x]K is the free K vector on f(x), Ωk[x]⊗k[x]K is the free K vectorspace on dx and δ takes f to df ⊗ 1 = f ′(x)dx ⊗ 1, which maps to f ′(ζ)dx underthe natural isomorphism

Ωk[x]/k ⊗k[x] K → K⟨dx⟩g(x)dx⊗ a 7→ ag(ζ)dx,

so δ is a map between 1-dimensional K vector spaces which is an isomorphism if ′(ζ) = 0. Since f is the minimal polynomial of ζ, f ′(ζ) is zero i ζ is not arepeated root of its minimal polynomial (i.e. k → K = k(ζ) is separable). Thisproves that (2), (4), and (5) are equivalent (for primitive extensions), and clearly(3) implies (4). But (4) and (2) together are equivalent to (3) for any ring map atall (Corollary 5.1.4) so the lemma is proven for the case of a primitive extension.

In general, the Primitive Element Theorem ensures that any nite separable ex-tension is primitive, so (5) implies all the other conditions by what we alreadyproved. One could avoid quoting the primitive element theorem here: To prove (5)implies (3) (hence also (2) and (4)), just factor k → K as a nite tower of primitiveseparable extensions

k = k0 → k1 → k2 → · · · → kn = K(5.7.2)

and note that a composition of formally étale maps is formally étale (Proposi-tion 5.1.1).

It remains to prove that if k → K is inseparable, then k → K is not (formally)smooth and ΩK/k = 0. Factor k → K as a nite chain of nite primitive extensionsas in (5.7.2). Choosing primitive elements ζi ∈ ki for each extension ki−1 → kiand presenting ki as ki−1[x]/fi(x) (with fi(x) the minimal polynomial of ζi overki−1), we get a presentation K = k[x1, . . . , xn]/(f1(x1), . . . , fn(xn)) of K as a k-algebra. Fix an algebraic closure k → k and let K := k ⊗k K. If k → K wereformally smooth, then the base change k → K would also be formally smooth(Proposition 5.1.1). Similarly, ΩK/k is zero i ΩK/k = ΩK/k ⊗K K = 0 becausek → k and K → K are faithfully at. But after passing to an algebraic closure, thefi(xi) all factor completely and at least one has a repeated root (otherwise k → Kwould be separable), so K is a k-algebra which is nite as an k-vector space, but


with a nontrivial nilpotent and one can see directly from the denition that such ank-algebra cannot be formally smooth and cannot have ΩK/k = 0 (Exercise 10). Denition 14. A eld k is called perfect i any nite eld extension k → K isseparable.

For example, any eld of characteristic zero is perfect, any algebraically closedeld is perfect, and any nite eld is perfect. If k has characteristic p > 0, then k(x)is not perfect because k(x) → k(x1/p) is a nite inseparable extension.

Theorem 5.7.2. If k is a perfect eld, then every eld extension k → K is separableand a k-algebra is separable i it is reduced.

Proof. [Mat2, Theorem 26.3]. The proof is not hard; the key point is to show that anitely generated (but not necessarily algebraic) eld extension k → K is separablei K ⊗k k

1/p is reduced.

In particular, the above theorem implies that a k-algebra A is separable i it isgeometrically reduced" in the sense that A⊗k k is reduced for an algebraic closurek of k.

The proof of the key point" in the proof of Theorem 5.7.2 in fact shows that anynitely generated separable eld extension k → K can be factored as

k → L → K

where k → L = k(x1, . . . , xn) is nitely generated purely transcendental and L → Kis nite separable (this statement is [Mat, 27.F, Lemma 3]). The purely transcen-dental extension k → L is a localization of a polynomial ring k[x1, . . . , xn] over k, soit is formally smooth over k and ΩL/k is the L-vector space with basis dx1, . . . , dxn.The nite separable extension L → K is étale (Lemma 5.7.1), so the compositionk → K is also formally smooth (Proposition 5.1.1(1)) and

0→ ΩL/k ⊗L K → ΩK/k → ΩK/L → 0

is exact (Proposition 5.3.5). But ΩK/L = 0 by Lemma 5.7.1, so ΩK/k is the K vectorspace with basis dx1, . . . , dxn. This proves:

Lemma 5.7.3. Any nitely generated separable eld extension k → K is formallysmooth and ΩK/k has dimension equal to the transcendence degree of K over k.

In fact we don't need the nite generation to conclude formal smoothness inLemma 5.7.3:

Theorem 5.7.4. (Cohen) A eld extension k → K is formally smooth i it isseparable.

Proof. [Mat, 28.L, Theorem 62] or [EGA, IV0.19.6.1] (the proof is basically the samein both sources, it just depends whether one wants it in French or English). Theeasy" direction is to prove that formally smooth implies separable: Suppose, to thecontrary, that k → K is formally smooth, but not separable. Since k → K is not


separable, there is a nitely generated eld extension k → L such that A := L⊗kKis not reduced (Remark 17). Factor k → L as a purely transcendental extensionk → F followed by a nite extension F → L. Note that K ′ := K ⊗k F is a eldit is just a purely transcendental extension of K. Since K ′ → A is a pushout ofF → L, the K ′-algebra A is nite dimensional as a K ′-vector space, so A must bean artinian ring. On the other hand, A is formally smooth over the eld L becauseformally smooth ring maps are stable under pushout (Proposition 5.1.1(2)), so itmust be reduced (Exercise 10), a contradiction.

The basic point for the hard" direction is to reduce to the nitely generated casediscussed above by limit arguments. To do this one needs some mechanism thatcommutes with ltered direct limits and can be used to check formal smoothnessin the aforementioned references they use Hochschild Homology. This might seem alittle subtle because in general a ltered direct limit of formally smooth ring mapsis not generally formally smooth (Example 2), but this diculty can't arise whenthe limit is a eld because all modules over a eld are projective.

Instead of using Hochschild Homology, let us prove the hard" direction usingthe theory of the truncated cotangent complex developed in 6. First of all, ourseparable extension k → K is the ltered direct limit of the nitely generatedsubextensions k → L of k → K. Each of these k → L must also be separable sinceL ⊆ K, so L ⊗k F ⊆ K ⊗k F for any eld extension k → F and a subring of areduced ring is reduced. By the nitely generated case of Lemma 5.7.3, each k → Lis formally smooth, hence H1(LL/k) = 0 (Corollary 6.4.3). Since the cotangentcomplex commutes with ltered direct limits (as does cohomology), LK/k is theltered direct limit of the LL/k, so we have H1(LK/k) = 0. Note that ΩK/k isautomatically projective since K is a eld, so we have

Ext1K(LK/k, V ) = Ext1K(ΩK/k, V )

= 0

(using Lemma 13.8.3 and the fact that H0(LK/k) = ΩK/k for the rst equality)for each K-vector space V . But then the Fundamental Theorem of the CotangentComplex (Theorem 7.2.1) implies that every k-algebra extension of K is trivial,hence k → K is formally smooth (Lemma 5.1.2).

Corollary 5.7.5. If k is a perfect eld, then any eld extension k → K is formallysmooth.

Proof. Combine the previous two theorems.

5.8. The Jacobian criterion. The purpose of this section is to establish the Ja-cobian Criterion for smoothness of a nite type algebra over a perfect eld (Deni-tion 14, 5.7). This criterion is often formulated over an algebraically closed eld,but it does not take much more work to do it over a perfect eld. We will make useof several commutative algebra results from 4.


Theorem 5.8.1. Let k be a perfect eld, A a smooth k-algebra, A→ B a surjectionof k-algebras with kernel I. For x ∈ SpecB with preimage y ∈ SpecA, the followingare equivalent:

(1) Bx is formally smooth over k.(2) The sequence 0→ Iy/I

2y → ΩAy/k ⊗Ay Bx → ΩBx/k → 0 is split exact.

(3) The sequence 0 → Iy/I2y → ΩAy/k ⊗Ay Bx → ΩBx/k → 0 is exact and ΩBx/k

is a free Bx-module (of nite rank).(4) I ⊗A k(x)→ ΩA/k ⊗A k(x) is injective.(5) There is an f ∈ B \ x such that Bf is smooth over k.(6) Bx is a regular local ring.

If x is a maximal ideal, these conditions are also equivalent to:

(7) The dimension of the cokernel of I ⊗A k(x) → ΩA/k ⊗A k(x) is equal to theKrull dimension of Bx.

The set of x ∈ SpecB satisfying these equivalent conditions is open in SpecB. Inparticular, if these conditions are satised for every maximal ideal x of B, then Bis smooth over k.

Proof. Since A → B is surjective and A is nite type over k (by denition ofsmooth"), B is also nite type (hence also of nite presentation) over k and Iis a nitely generated (hence nitely presented) A-module, hence I/I2 = I⊗AB is anitely generated B-module. Since k → A is smooth, ΩA/k is a locally free A-moduleof nite rank (Proposition 5.5.3), so it is clear that (2) and (3) are equivalent. Therst ve conditions are hence equivalent by Theorem 5.6.1 and it is clear that (5) isan open condition.

Note that k(x) = k(y) since A → B is surjective. Let t be the transcendencedegree of k(x) = k(y) over k. Since k is assumed perfect, the nitely generated eldextension k → k(x) is automatically separable (Theorem 5.7.2), hence it is formallysmooth and Ωk(x)/k has dimension t as a k(x) vector space (Lemma 5.7.3). Sincek → k(x) is formally smooth, we have an exact sequence of k(x) vector spaces

0→ mx/mx → ΩBx/k ⊗Bx k(x)→ Ωk(x)/k → 0(5.8.1)

associated to the ring maps k → Bx → k(x) (Proposition 5.3.4). Note that

ΩBx/k = ΩB/k ⊗B k(x),

so exactness of (5.8.1) yields

t+ dimk(x) mx/m2x = dimk(x)ΩB/k ⊗B k(x).(5.8.2)

Looking at the corresponding sequence for k → Ay → k(y) = k(x), we nd

t+ dimk(x)my/m2y = dimk(x) ΩA/k ⊗A k(x).(5.8.3)

We will now prove the equivalence of (4) and (6) under the assumption that eachlocal ring Ay is regular. Before we prove this, we note that once it has been proved,we get the result in general by showing that the assumption that Ay is regular is


automatically implied by the assumption that A is smooth over k. For the latterassertion, we can just apply the known case to a presentation k[x1, . . . , xn] → Aof A because we do know that the local rings of k[x1, . . . , xn] are regular (Proposi-tion 4.3.2).

Under assumption (4),

0→ I ⊗A k(x)→ ΩA/k ⊗A k(x)→ ΩB/k ⊗B k(x)→ 0(5.8.4)

is exact. Note that I ⊗A k(x) = Iy ⊗Ay k(y) and Iy is the kernel of the surjectionAy → Bx because localization is exact. Set d := dimk(x) I ⊗A k(x). By Nakayama'sLemma, Iy can be generated by d elements. Since Ay is regular, we have

dimAy = dimk(x) my/m2y,(5.8.5)

where dimAy is the Krull dimension of Ay. Putting (5.8.5) together with the exact-ness of (4) and formulas (5.8.2) and (5.8.3), we nd:

dimk(x)mx/m2x = dimAy − d.

But since Iy can be generated by d elements and Bx = Ay/Iy, the right hand side isat most dimBx by the Hauptidealsatz. On the other hand, the left hand side is atleast dimBx because we have

dimBx ≤ dimk(x)mx/m2x

for any noetherian local ring (by the Hauptidealsatz and Nakayama). We concludethat dimBx = dimk(x)mx/m

2x, so Bx is regular.

Under the assumption (6) that Bx is regular, Ay → Bx is a surjection of regularlocal rings, hence its kernel Iy can be generated by a regular sequence of length

d := dimAy − dimBx

(Lemma 4.3.7), hence Iy/I2y is a free Bx-module of rank d (Lemma 4.2.1), so we have

dimk(x) I ⊗A k(x) = dimk(x) Iy ⊗Ay k(x)(5.8.6)= dimk(x) Iy ⊗Ay Bx ⊗Bx k(x)

= dimk(x) Iy/I2y ⊗Bx k(x)

= d.

Since Bx is regular, we have

dimBx = dimk(x)mx/m2x.(5.8.7)

From (5.8.5), (5.8.7), (5.8.2), and (5.8.3), we nd

dimk(x)ΩA/k ⊗A k(x)− dimk(x) ΩB/k ⊗B k(x) = d.(5.8.8)

By the dimension formulas (5.8.6) and (5.8.8), the left map in the exact sequence

I ⊗A k(x)→ ΩA/k ⊗A k(x)→ ΩB/k ⊗B k(x)→ 0(5.8.9)

must be injective on dimension grounds, which proves (4).


When x is maximal, k(x) is a nite extension of k by the Nullstellensatz ([Eis,4.19], for example), so t = 0 and (5.8.1) becomes

dimk(x)mx/m2x = dimk(x)ΩB/k ⊗B k(x).(5.8.10)

The usual exact sequence (5.8.9) hence says that the dimension of the cokernel ofI ⊗A k(x)→ ΩA/k ⊗A k(x) is equal to dimk(x)mx/m

2x, so it is clear that (6) and (7)

are equivalent. Corollary 5.8.2. Let X be a scheme of locally nite type over a perfect eld k. Fora point x ∈ X, the following are equivalent:

(1) There is a neighborhood U of x in X such that U is smooth over k.(2) OX,x is formally smooth over k.(3) OX,x is a regular local ring.

The set of x ∈ X satisfying these equivalent conditions is open and X is smoothover k i these conditions are satised by every x (or equivalently, every closed x).

Proof. This is all local in nature, so we can reduce to the case where X = SpecBis ane, then we can choose some k-algebra surjection A = k[x1, . . . , xn] → B andapply the theorem.

5.9. Artinian lifting property. Although formal smoothness is formulated interms of a general lifting property with respect to all nilpotent surjections of ringsT ′ → T , it is desirable in applications to have a slightly easier condition to check(under some hypotheses on the map in question).

Denition 15. Let f : A → B be a local map of local rings. We say that f hasthe artinian lifting property i there is a completion as indicated in any diagram oflocal maps of local rings

T Boo

~~T ′

i

OO

Aoo

f

OO(5.9.1)

where i : T ′ → T is a small thickening (Denition 5) and B → T induces anisomorphism on residue elds.

Remark 18. In light of Lemma 4.1.2, one can replace small thickening" withsurjective local map of local artinian rings" without changing the meaning of theartinian lifting property.

Lemma 5.9.1. A local map of local rings f : A → B has the artinian liftingproperty i the induced map f : A → B of complete local rings has the artinianlifting property.

Proof. The maximal ideal of a local artinian ring is nilpotent (Theorem 4.1.1), soin any solid diagram (5.9.1) as in Denition 15, the horizontal arrows will kill somelarge enough power of the maximal ideals of A and B, so (5.9.1) will factor through


a similar solid diagram with f replaced by the map A/mnA → B/mn

B induced by f .Similarly, any lifting B → T ′ must factor through the quotient map B → B/mn

B

(for a possibly larger n). But these quotients are the same as the correspondingquotients for the completions.

Lemma 5.9.2. Let k be a eld and let (A,m) be a noetherian local k-algebra whoseresidue eld K = A/m is a formally smooth extension of k (c.f. 5.7). Then thefollowing are equivalent:

(1) A is regular.

(2) A is regular.(3) k → A has the artinian lifting property.

(4) k → A has the artinian lifting property.

(5) A is isomorphic to K[[x1, . . . , xn]] as a k-algebra.

Proof. (C.f. [H2, 4.6]) (1) and (2) are equivalent by Proposition 4.3.3 and (3) and(4) are equivalent by Lemma 5.9.1. It is easy to see directly from the denition ofthe artinian lifting property that (5) implies (4). Clearly (5) implies (2).

The rst nontrivial statement is that (2) implies (5), which is part of the generalCohen structure theory [Mat, 28.M]; this is proved as follows: We rst use theassumption that k → K is formally smooth to successively lift in the diagram ofk-algebras below.

K

uu=

· · · // A/m3 // A/m2 // A/m

Since A is, by denition, the inverse limit of the A/mn, we get a k-algebra sectionK → A of the natural projection A→ A/m = K. Pick a minimal set of generatorsm1, . . . ,mn for m (equivalently, by Nakayama, pick m1, . . . ,mn ∈ m whose imagesin m/m2 form a K-basis). Set B := K[[x1, . . . , xn]] and let n = (x1, . . . , xn) bethe maximal ideal of B. Since A is complete, we have a K-algebra map (whichis a fortiori a k-algebra map) f : B → A taking xi to mi. Using the criterion ofTheorem 4.3.1(8) for the regularity of A, it is straighforward to check by inductionon n that the maps fn : B/nn → A/mn are isomorphisms by checking that theyare isomorphisms of K vector spaces. Since f is the inverse limit of the fn, it is anisomorphism.

It remains only to prove that (4) implies (5). As above, we start by using theassumption on k → K to produce a k-algebra map K → A. Let m1, . . . ,mn, B,m, n and f : B → A be as above. We want to use (4) to prove that f is anisomorphism. First of all, f is surjective (because it is elementary to check that itis surjective modulo any given power of the maximal idealsc.f. the proof of [Mat,28.J, Corollary 1]), even without assuming (4). Next, the induced surjective mapf2 : B/n2 → A/m2 is also easily seen to be an isomorphism by noting that it is a


map of K vector spaces, both of dimension n+1. Now we use (4) (plus Remark 18)to lift successively in the diagram below.

A

uu

mi 7→xi

· · · // B/n4 // B/n3 // B/n2

(One factors B/nm+1 → B/nm into a sequence of small extensions by quotientingout by one degree m monomial at a timethe kernel of the map in question isthe K-vector space spanned by the degree m monomials in the xi. Then one canuse the hypothesized lifting (4) to lift in the resulting ner" diagram.) Since B isthe inverse limit of the bottom row, we thus obtain a map g : A → B where thecomposition gf : B → B is the identity mod n2, so it takes xi to xi +O(2). I claimthat this implies gf is injective. The claim implies that f must be injective, henceit must be an isomorphism because it is surjective. To prove the claim, considera typical non-zero element p =

∑I aIx

I of B and let us prove that gf(p) = 0.For an exponent vector I = (i1, . . . , in), set |I| := i1 + · · · + in and let N be theminimum of all the |I| for which aI = 0. Since gf(xi) = xi + O(2), we then havegf(p) =

∑|I|=N aIx

I +O(N +1), which is certainly non-zero since the coecient ofone of the degree N monomials must be non-zero. (In fact one can show that gf isan isomorphism in much the same way [Eis, 7.17].)

Note the similarity between the proof of Lemma 5.9.2 and the proof of Lemma 4.5.5.

Corollary 5.9.3. Let X be a scheme of locally nite type over a perfect eld k. Fora point x ∈ X, the following are equivalent:

(1) There is a neighborhood U of x in X so that U is smooth over k.(2) OX,x is formally smooth over k.(3) OX,x is a regular local ring.(4) k → OX,x has the artinian lifting property.(5) There is a lift in every diagram of schemes

SpecT //

X

SpecT ′ //

99

Spec k

where T ′ → T is a small thickening (Denition 5) and the unique point ofSpecT maps to x and induces an isomorphism on residue elds.

The set of x ∈ X satisfying these conditions is open in X. In particular X is smoothover k i these conditions hold for each closed point of X.

Proof. The rst three conditions are equivalent by Corollary 5.8.2. Conditions (3)and (4) are equivalent by Lemma 5.9.2 (applied with A = OX,xnote that thehypothesis on residue elds in Lemma 5.9.2 is satised in light of Corollary 5.7.5


because we assume k is perfect). Condition (5) is just a reformulation of (4) in moregeometric terms.

In fact, the above corollary is true in much greater generality, though the proof issignicantly more involved:

Theorem 5.9.4. Let f : X → Y be a locally nite type morphism of locally noether-ian schemes. For a point x ∈ X with image y := f(x), the following are equivalent:

(1) There is a neighborhood U of x in X so that f |U : U → Y is smooth.(2) OX,x is formally smooth over OY,y.(3) There is a lift in every diagram of schemes

SpecT //

X

f

SpecT ′ //

;;

Y

where T ′ → T is a small thickening (Denition 5) and the unique point ofSpecT maps to x and induces an isomorphism on residue elds.

The set of x ∈ X satisfying these conditions is open in X.

Proof. See [EGA, IV.17.5.4] and the discussion below.

I will provide a rough sketch of the proof of Theorem 5.9.4 which may be useful tothe reader who wants to go through the details in [EGA]. The uninterested reader isstrongly urged to move on to the next section. This is one place where it seems to bedicult to avoid some discussion of topologies in the theory of formal smoothness;the essential diculty is that one wants to make use of m-adic completion at variouspoints, but completion destroys (essential) niteness, so one needs some mechanismto remember" the niteness. For the sake of this discussion, the following denitionwill be general enough, though it is nowhere near the generality of [EGA]:

Denition 16. A local map of local rings f : A → B is called adically formallysmooth i there is a lift as indicated in any diagram of rings

T Boo

~~T ′

OO

Aoo

OO

where T ′ → T is surjective with square-zero kernel and B → T kills some power ofthe maximal ideal of B.

Remark 19. The condition that B → T kills, say, mnB implies that A→ T ′ must kill

m2nA since A→ T ′ must take mn

A into I = Ker(T ′ → T ) by commutativity and I2 = 0.For similar reasons, a lift must also kill m2n

B . Thus the solid diagram is the samething as a diagram of continuous maps of topological rings when A and B are giventhe m-adic topologies (one could also give A the discrete topology without changing


anything) and T and T ′ are given the discrete topology (any lift is automaticallycontinuous). The notion of adically formally smooth" in Denition 16 is the sameas Grothendieck's formellement lisse pour la topologie m-préadique .

The key feature of this denition is:

Lemma 5.9.5. A local map of local rings f : A→ B is adically formally smooth ithe induced map on completions f : A→ B is adically formally smooth.

Proof. The proof is the same as that of Lemma 5.9.1.

The rst step in the proof of Theorem 5.9.4 is relatively easy:

Lemma 5.9.6. Let f : A → C be a map of noetherian local rings of essentiallynite presentation. Then f is formally smooth i f is adically formally smooth.

Proof. The implication =⇒ is obvious from the denitions. For the other implica-tion, factor A → C as A → B → C, where B is a localization of A[x1, . . . , xn] at aprime lying over mA and B → C is a surjective of local rings with nitely generatedkernel I. By Theorem 5.4.1, f is formally smooth i the usual sequence

0→ I/I2 → ΩB/A ⊗B C → ΩC/A → 0

is split exact (we just need to check that the left map is injective and ΩC/A is at, forthen ΩC/A is free since it is nitely generated over the local ring C). The modules inthis sequence are nitely generated and C is noetherian, so we can check exactnesstopologically" (after completion), and this latter exactness follows from adic formalsmoothness in much the same way that the usual exactness follows from formalsmoothness of A→ C. C.f. [EGA, IV0.22.6.4].

The next step is much harder:

Theorem 5.9.7. Let f : A → B be a local map of local noetherian rings. Then fhas the artinian lifting property i f is adically formally smooth.

Proof. This is [EGA, IV0.22.1.4]. The only diculty is =⇒ . Notice that there is noessentially nite presentation" assumption, and, indeed, it is not clear that such anassumption would make the proof any easier: Grothendieck's proof uses completionin an essential way, and this will destroy any essential niteness hypotheses anyway.The proof proceeds in several steps:

Step 1. Reduction to these case where A and B are complete. This is immediatefrom Lemma 5.9.5 and Lemma 5.9.1. Note that we just destroyed any essentialniteness even if we assumed we had it to begin with.

Step 2. We handle the case where k is a eld. Our Lemma 5.9.2 is a pretty goodapproximation to this. The general proof roughly reduces to Lemma 5.9.2 by makinguse of the prime subeld k0 ⊆ k (which is perfect) and the fact that k0 → K = B/mB

is formally smooth (Corollary 5.7.5). This step is fairly elementary and is explainedin [EGA, IV0.22.1.3].


Step 3. Set B := B ⊗A k = B/mAB, where k := A/mA. Using stability underbase change," we check that the artinian lifting property for f implies the artinianlifting property for k → B, hence k → B is adically formally smooth by the resultof the previous step.

Step 4. Now it gets hairy. From the previous step, we know k → B is adicallyformally smooth. By some delicate existence" results [EGA, IV0.19.7.1-2], we cannd (using the fact that A is noetherian and B is complete) a complete, localnoetherian ring B′, a at, local map A → B′, and an isomorphism of k-algebrasu0 : B → B

′:= B′ ⊗A k.

Step 5. Finally, we argue by a jazzed up version of the proof of Lemma 5.9.2, thatwe can lift u0 to an isomorphism of local rings u : B → B′ over A. Since B and B′

are complete, we can do this fairly easily by inductively constructing isomorphismsun : B/mn+1

B → B′/mn+1B′ , using the artinian lifting property to build un+1 from

un.

5.10. Characterization of smoothness. The following result is the culminationof the theory of formal smoothness:

Theorem 5.10.1. Let f : X → Y be a morphism of schemes of locally nitepresentation, x ∈ X, y := f(x). The following are equivalent:

(1) f is smooth on a neighborhood of x.(2) f is at at x and the ber Xy = f−1(y) is smooth over Spec k(y) near x.(3) f is at at x and the geometric ber Xy is regular at x.(4) The ring homomorphism OY,f(x) → OX,x is formally smooth.

The set of x ∈ X satisfying these equivalent conditions is open in X.

Remark 20. In Theorem 7.5.1 of 7.5 we will give a characterization of (formal)smoothness in terms of the cotangent complex (the subject of ß6-8).

Proof. This is [EGA, IV.17.5.1]. We proved the equivalence of (1) and (4) in Corol-lary 5.6.3. To see that these equivalent conditions imply (2), we note that the onlyissue is the atness (smoothness is stable under base change)we will prove this inTheorem 5.10.3 below. We will leave it to the reader to look at [EGA] for the otherequivalences.

For the sake of completeness we also mention the analogous characterization ofétale maps:

Theorem 5.10.2. Let f : X → Y be a morphism of schemes of locally nitepresentation, x ∈ X, y := f(x). The following are equivalent:

(1) f is étale on a neighborhood of x.(2) f is at at x and unramied on a neighborhood of x.(3) f is at at x and OX,x/myOX,x is a eld, nite and separable over k(y).(4) OY,f(x) → OX,x is formally étale.


The set of x ∈ X satisfying these equivalent conditions is open in X.

Proof. [EGA, IV.17.6.1] Theorem 5.10.3. Let f : A → C be a formally smooth local map of local rings ofessentially nite presentation. Then C is at over A.

Proof. Since A → C is of essentially nite presentation, it can be factored as asequence of local maps of local rings A→ B → C where B is the localization of thepolynomial ring A[x1, . . . , xm] (at some prime ideal lying over mA) and B → C issurjective with nitely generated kernel I ⊆ mB. Let k be the residue eld of A andlet B := B ⊗A k, C := C ⊗A k. We have a pushout diagram of local maps of localrings

A //

B //

C

k // B // C

where the vertical arrows are surjective. Since B → C is is also surjective, B, C, B,and C have the same residue eldcall it K. Set

I := I ⊗B B = I ⊗A k = I/mAI.

Warning: We do not know (yet) that the natural map I → B onto Ker(B → C)is injective because we do not know (yet) that C is at over A.

Since A→ C is formally smooth, the natural sequence

0→ I/I2 → ΩB/A ⊗B C → ΩC/A → 0(5.10.1)

is exact and ΩC/A is projective (Theorem 5.4.1). In particular ΩC/A is at, so (5.10.1)stays exact after applying _ ⊗C K. Since

I/I2 ⊗C K = I ⊗B C ⊗C K

= I ⊗B K

= I ⊗B B ⊗B K

= I ⊗B K

and

ΩB/A ⊗B C ⊗C K = ΩB/A ⊗B K

= ΩB/A ⊗B B ⊗B K

= ΩB/k ⊗B K,

and ΩC/A ⊗C K = ΩC/k ⊗C K, the resulting exact sequence can be written

0→ I ⊗B K → ΩB/k ⊗B K → ΩC/k ⊗C K → 0.(5.10.2)

Pick b1, . . . , bn ∈ I ⊆ mB so that the images of the bi in

I ⊗B K = I ⊗B K


form a K-basis. Then b1, . . . , bn generate I by Nakayama's Lemma, so their imagesb1, . . . , bn ∈ I = I/mAI also generate the B-module I.

Note that ΩB/k is the free B-module on dx1, . . . , dxm and the left map in (5.10.2)takes bj (or rather, its image in I ⊗B K) to dbj =

∑mi=1(∂bj/∂xi)dxi (note that the

bj live in some localization B of the polynomial ring k[x1, . . . , xm], so they are justratios of polynomials in the xi). The fact that the left map in (5.10.2) is injectivemeans that the Jacobian matrix of partial derivatives

J := (∂bj/∂xi)1≤j≤n1≤i≤m ∈ Matm×n(B)

has rank n when viewed as a map of K vector spaces by reducing modulo mB (thematrix J mod mB is just the matrix for the linear transformation

d : I ⊗B K → ΩB/k ⊗B K

appearing on the left of (5.10.2) using the bases b1, . . . , bn and dx1, . . . , dxm). Bylinear algebra, this means that n ≤ m and some n× n minor of J is invertible modmB. After possibly reordering the xi, we can assume it is the upper left minor, sothe matrix

J ′ := (∂bj/∂xi)1≤j≤n1≤i≤n ∈ Matn×n(B)

is invertible mod mB.

I claim that b1, . . . , bn ∈ mB are linearly independent modulo m2B. Suppose not.

Then there are w1, . . . , wn ∈ B, not all in mB, such that∑n

j=1 wjbj ∈ m2B. Taking

partial derivatives with respect to xj, we nd that∑m

j=1wj(∂bj/∂xi) ∈ mB for j =1, . . . , n. But reading this mod mB then says that some nontrivial linear combinationof the columns of J ′ is zero mod mB, which contradicts the fact that J ′ is invertiblemod mB.

Since B is a localization of the polynomial ring k[x1, . . . , xm], it is a regular localring. Hence, by Lemma 4.3.6, the claim above implies that b1, . . . , bn is a regularsequence in B. Note that B is certainly at over A and essentially of nite presen-tation (it is the localization of a polynomial ring over A), so Lemma 4.5.4 impliesthat b1, . . . , bn is a regular sequence in B and that

B/(b1, . . . , bn) = B/I = C

is at over A.

6. The Truncated Cotangent Complex

In this section, we are going to construct (our version of) the truncated cotangentcomplex LB/A of a morphism of sheaves of rings A → B on a topological space.Until further notice, we will refer to LB/A simply as the cotangent complex since wedo not yet have any untruncated" cotangent complex. The complex LB/A is a twoterm complex

LB/A = [L1 → L0]


of B-modules functorially associated to A → B. It is equipped with a naturalmap LB/A → ΩB/A inducing an isomorphism H0(LB/A) = ΩB/A (again natural inA → B) and can therefore be viewed as a renement of the B-module ΩB/A ofKähler dierentials (3).

We use chain complexes as opposed to cochain complexes so our dierential L1 →L0 decreases degree by 1 and our complex is sitting in degrees 1, 0. The readerwho prefers cochain complexes can always convert to that language using the usualconvention Cn := C−n, Hn := H−n, so our complex LB/A could be viewed as acochain complex sitting in degrees −1, 0. (I prefer to avoid negative numbers.)

The complex LB/A is constructed as follows: We forget the ring structure on B,regard it as a sheaf of sets, and let A[B] be the free A-algebra on that sheaf of sets(the sheaed version of the polynomial ring over A in variables xb, one for eachelement b ∈ B). Then we have a tautological surjection of A-algebras A[B] → B(the sheaed version of xb 7→ b). We let I be its kernel and we set

LB/A := [I/I2 → ΩA[B]/A ⊗A[B] B],

where the boundary map for this complex and the natural map LB/A → ΩB/A

inducing an isomorphism on H0 are from the exact sequence of Corollary 3.7.3 of3.7.

The A-algebra A[B] is rather large," hence the actual complex LB/A is somewhathorrendous, but as an object of the derived category of B-modules it is not so bad."For example, we will see that if R is a stalkwise formally smooth A-algebra (andmore generally...), then a surjection R → B of A-algebras with kernel J yields anatural quasi-isomorphism

LB/A = [J/J2 → ΩR/A ⊗R B].

In particular, if B itself is stalkwise formally smooth (and more generally), then thenatural map LB/A → ΩB/A is a quasi-isomorphism.

6.1. Flat modules. Let A be a sheaf of rings on a space X. Recall that an A-module M is called at i the right exact functor

_ ⊗A M : Mod(A)→Mod(A)

is exact. Recall thatM⊗AN is the sheaf associated to the presheaf U 7→M(U)⊗A(U)

N(U). Tensor products commute with ltered direct limits, so the stalks of M⊗ANare given by

(M ⊗A N)x = Mx ⊗Ax Nx.

Since exactness can be checked on stalks, it follows that if each Mx is a at Ax-module, then M is a at A-module. The converse also holds (Exercise 9). Inparticular, if M is stalkwise free (each stalk Mx is a free Ax-module), then M is at.

There are lots of at (even stalkwise free) A-modules! If F is any sheaf (of sets)on X, then one can form the free module ⊕FA on F . By denition, this is the sheaf


associated to the presheafU 7→ ⊕F (U)A(U).

Here of course ⊕F (U)A(U) means the free A(U)-module on the set F (U). Therestriction maps for this presheaf are dened using the fact that formation of thefree A-module on a set S is a functor:

An× Sets → AnMod

(A, S) 7→ (A,⊕SA)

This functor preserves ltered direct limits: If (Ai, Si) is a ltered direct limit sys-tem in An × Sets with limit (A, S), then (A,⊕SA) is the ltered direct limit of(Ai,⊕Si

Ai) in AnMod.

This preservation of ltered direct limits ensures that the stalk of ⊕FA at a pointx is given by

(⊕FA)x = ⊕FxAx

(in particular this implies ⊕FA is a stalkwise free, hence at A-module) and, moregenerally, that formation of ⊕AF commutes with change of topological space": Iff : X → Y is a map of spaces and A (resp. F ) is a sheaf of rings (resp. sets) on Y ,then we have a natural isomorphism of f−1A modules

f−1(⊕FA) = ⊕f−1Ff−1A.

Compare the discussion of dierentials in 3.2.

Given any A-module M , we can forget that M is an A-module and just take thefree A-module ⊕MA on the underlying sheaf of sets. There is a natural surjection ofA-modules ⊕MA → M , functorial in Mnamely the image of Id : M → M underthe adjunction isomorphism

HomSh(X)(M,M) → HomMod(A)(⊕MA,M).

6.2. Symmetric algebra. Let A be a ring, M an A-module. The A-algebraSym∗AM , called the symmetric algebra over A on M is, by denition, the quotientof the tensor A-algebra ⊕∞n=0M

⊗n (which is non-commutative) by the two-sidedideal generated by the degree one terms of the form m⊗ n− n⊗m for m,n ∈ M .Formation of Sym∗AM denes a functor

Sym∗ : AnMod → FlAn

(A,M) 7→ (A→ Sym∗AM)

which is left adjoint to the forgetful functor

FlAn → AnMod

(A→ B) 7→ (A,B).

The functor Sym∗ preserves ltered direct limits.


If A is now a sheaf of rings on a space X and M an A-module, then the symmetricalgebra Sym∗AM on M is the A-algebra associated to the presheaf

U 7→ Sym∗A(U)M(U).

Preservation of ltered direct limits ensures that the stalks of Sym∗AM are given by

(Sym∗AM)x = Sym∗AxMx

and that formation of Sym∗AM commutes with change of topological spaces.

The symmetric algebra Sym∗AM comes with an A-module direct sum decomposi-tion

Sym∗A M = ⊕∞n=0 SymnAM

= A⊕M ⊕ Sym2 M ⊕ · · ·making it a graded A-algebra. In particular, we have a natural inclusion of A-modules M → Sym∗A M (the isomorphism onto the degree one part) which we oftensuppress in the notation, simply regarding local sections of M as local sections ofSym∗AM via this inclusion. This sheaed" symmetric algebra functor is left adjointthe forgetful functor FlAn(X)→ AnMod(X). In particular, the symmetric algebrafunctor Sym∗A takes direct limits of A-modules to direct limits of A-algebras, and,in particular, this is true for nite direct sums:

Sym∗A(M ⊕N) = (Sym∗A M)⊗A (Sym∗AN).(6.2.1)

If F is a sheaf of sets on X and A is a sheaf of rings, then the symmetric algebraover A on the free A-module ⊕FA will be denoted A[F ]. Formation of A[F ] isfunctorial in F (and A), and F 7→ A[F ] is left adjoint to the forgetful functor fromA-algebras to sheaves of sets on X. The stalks of A[F ] are given by

A[F ] = Ax[Fx],

where Ax[Fx] is the polynomial ring over Ax in variables xf : f ∈ Fx and formationof A[F ] commutes with change of topological space:

f−1(A[F ]) = (f−1[A])(f−1F ).

There is a natural map of S := Sym∗A M -modules

M ⊗A S → ΩS/A(6.2.2)m⊗ s 7→ sdm.

Lemma 6.2.1. For any A-moduleM , the natural map (6.2.2) yields an isomorphismΩS/A = M⊗AS. In particular, if M is a at A-module, then ΩS/A is a at S-module.Similarly, if M is free with basis xi, then ΩS/A is free with basis dxi.

Proof. Let N be an arbitrary S-module. Suppose D : S → N is an A-linear deriva-tion (3.5). Restricting D to the degree one part S1 = M , we obtain a map ofA-modules D1 : M → N (regarding N as an A-module via restriction of scalarsalong A → S). The map D is uniquely determined by D1 because of the Leib-nitz Rule and the fact that S is generated as an A-algebra by M . On the other


hand, given any A-linear map f : M → N we can construct an A-linear derivationD : S → N with D1 = f by setting

D(m1 ⊗ · · · ⊗mn) :=n∑

i=1

f(mi)m1 ⊗ · · · ⊗mi−1 ⊗mi+1 ⊗ · · · ⊗mn

(this D is clearly A-linear and symmetric and it is easy to check the Leibnitz Rule).We have established a natural bijection

DerA(S,N) = HomMod(A)(M,N).

If we put this together with the natural isomorphism (3.5.2) of 3.5, then we nd

HomMod(S)(ΩS/A, N) = DerA(S,N)

= HomMod(A)(M,N)

= HomMod(S)(M ⊗A S,N),

which shows that ΩS/A and M ⊗A S are isomorphic S-modules since they have thesame maps to an arbitrary S-module N . If one follows through each of the abovenatural bijections, then one sees that this isomorphism is realized by our naturalmap (6.2.2).

In the next lemma we will use the following standard fact from commutativealgebra:

Theorem 6.2.2. (Lazard) For a ring A and an A-module M , the following areequivalent:

(1) M is at.(2) There is a ltered partially ordered set I and a direct limit system (Mi) of

nitely generated free A-modules indexed by I such that M is isomorphic tothe direct limit of the Mi.

Proof. See Lazard's original article [L], or [B, Chapter X, 1.6]. Lemma 6.2.3. Let A be a ring, M a at A-module. Then Sym∗AM is a ltereddirect limit of polynomial rings over A in nitely many variables. In particular it isa ltered direct limit of formally smooth A-algebras.

Proof. By Lazard's Theorem, M is a ltered direct limit of nitely generated freeA-modulesMi. Formation of Sym∗A commutes with ltered direct limits, so Sym∗AMis the ltered direct limit of the Sym∗AMi. If we choose a basis x1, . . . , xn for Mi,then Sym∗A Mi is identied with the polynomial ring A[x1, . . . , xn]. Example 2. A ltered direct limit of formally smooth algebras (even a ltereddirect limit of nite type polynomial rings) need not be formally smooth. Take anyring A and any A-module M which is at but not projective (e.g. A = Z, M = Q).Let S := Sym∗AM . Then S is a ltered direct limit of formally smooth A-algebrasby Lemma 6.2.3, but I claim S is not formally smooth over A. If it were, then ΩS/A

would be a projective S-module (Corollary 5.3.3). The structure map A→ S has a


retract S → A. Indeed, we obtain a diagram A→ S → A with composition IdA byapplying the functor Sym∗A to the diagram 0 → M → 0 of A-modules. If A → Bis any map of rings and M is a projective A-module, then A ⊗B M is a projectiveB-module because

HomB(A⊗B M, _ ) = HomA(M, _ ).

So in our situation, if ΩS/A were a projective S-module, then ΩS/A ⊗S A would bea projective A-module. But ΩS/A = M ⊗A S by Lemma 6.2.1, so ΩS/A ⊗S A = M ,which is not projective by assumption.

6.3. Functoriality of LB/A. Let A be a sheaf of rings on a space X and let B be anA-algebra. Recall from the beginning of 6 that we dened a complex of B-modules

LB/A := [I/I2 → ΩA[B]/A ⊗A[B] B]

(sitting in degrees −1, 0) called the cotangent complex of A→ B. Here A[B] is thefree A-algebra on (the sheaf of sets underlying) B and I is the kernel of the naturalsurjection A[B] → B. Recall that we have a natural map LB/A → ΩB/A inducingan isomorphism on H0 (by applying Corollary 3.7.3 to A→ A[B]→ B).

The complex LB/A is functorial in A→ B in the sense that a commutative diagramof rings (or sheaves of rings)

A //

B

A′ // B′

(6.3.1)

(i.e. a FlAn(X) morphism) gives a commutative diagram of sheaves of rings

A

// A[B] //

B

A′ // A′[B′] // B′

and hence a morphism of complexes of B-modules

[I/I2 //

ΩA[B]/A ⊗A[B] B]

[I ′/I

′2 // ΩA′[B′]/A′ ⊗A′[B′] B′]

or, equivalently, a map of complexes of B′-modules LB/A ⊗B B′ → LB′/A′ .

Formation of LB/A also commutes with change of space: If f : X → Y is a mapof topological spaces and A→ B is a map of sheaves of rings on Y , then we have anatural isomorphism

f−1LB/A = Lf−1B/f−1A(6.3.2)

of complexes of f−1B-modules. Indeed, one simply combines the compatibility ofdierentials, kernels, and free algebras with change of spaces.


Denition 17. For a map of ringed spaces f : X → Y , the cotangent complex off , denoted LX/Y is, by denition, the cotangent complex LOX/f−1OY

.

All functoriality and results for the cotangent complex of a map of sheaves ofrings on a xed space are easily translated into analogous results for the geometric"cotangent complexes of the above denition.

6.4. Comparison theorem. Because the cotangent complex LB/A is dened usingthe rather large" functorial factorization A→ A[B]→ B of a ring map A→ B, itis hardly ever a complex of nitely generated B-modules. However, we will show inTheorem 6.4.2 of this section that LB/A can be computed up to quasi-isomorphismby using various alternative factorizations A → S → B of A → B. In particular,this will allow us to show that the cohomology modules of LB/A are nitely generatedfor reasonable ring maps A→ B (Corollary 6.4.5).

Lemma 6.4.1. Let A → R → S → B be a diagram of rings. Suppose R → B issurjective with kernel J and S → B is surjective with kernel K. Then there is anatural map of complexes of B-modules (the rows below):

[J/J2 //

ΩR/A ⊗R B]

[K/K2 // ΩS/A ⊗S B]

This map is a quasi-isomorphism under either of the following assumptions:

(1) R → S is surjective and S is a ltered direct limit of formally smooth A-algebras.

(2) Both R and S are ltered direct limits of formally smooth A-algebras.

Variant: For such a diagram of sheaves of rings, the same conclusion holds whenat least one of the assumptions (1), (2) holds for each stalk of the diagram.

Proof. Suppose (1) holds. Let I be the kernel of the surjection R → S. By Propo-sition 5.3.4 our assumptions on S ensure that the sequence of S-modules

0→ I/I2 → ΩR/A ⊗R S → ΩS/A → 0(6.4.1)

(where the left map is given by i 7→ di ⊗ 1 as usual) is exact and ΩS/A is a atS-module, hence (6.4.1) stays exact after applying _ ⊗S B.


We have an exact diagram of R-modules:

0

0

0

0 // J2 ∩ I //

J2 //

K2 //

0

0 // I //

J //

K //

0

0 // I/(I ∩ J2) //

J/J2

// K/K2 //

0

0 0 0

Indeed, it is easy to see, using surjectivity of R → S, that the top two rows areexact, hence the bottom row is exact by the Snake Lemma. We have

(I/I2)⊗S B = (I ⊗R S)⊗S B

= I ⊗R B

= I/(IJ),

so when we tensor (6.4.1) over S with B the resulting sequence (which remainsexact!) takes the form

0→ I/(IJ)→ ΩR/A/JΩR/A → ΩS/A/KΩS/A → 0,

where the left map is i 7→ di. In general we only have a containment IJ ⊆ J2∩I, butthe very fact that I/(IJ) → ΩR/A/JΩR/A is injective means that this containmentmust be an equality: If it weren't, there would be j1, j2 ∈ J such that j1j2 ∈ I \ IJ ,so j1j2 would be nonzero in I/(IJ), but d(j1j2) = j1dj2 + j2dj1 is certainly zero inΩR/A/JΩR/A. So we have a commutative diagram with exact columns:

0

0

I/(J2 ∩ I)

I/(IJ)

J/J2 //

ΩR/A ⊗R B

K/K2 //

ΩS/A ⊗S B

0 0


We know our map of complexes induces an isomorphism on H0 even without theextra hypotheses on S (both H0's are ΩB/A by Corollary 3.7.3). The fact that ourmap of complexes induces an isomorphism on H1 follows from the Snake Lemma inthe above diagram.

Now suppose (2) holds. We are going to deduce that the indicated map is a quasi-isomorphism by reducing to the case established above. The tautological surjectionof R-algebras R[S]→ S sits in a commutative diagram

A→ R→ R[S]→ S → B

where the appropriate compositions are the maps from our original diagram. Let Ibe the kernel of the surjection f : R[S]→ B. Then we have a diagram of complexes

[J/J2 //

ΩR/A ⊗R B]

[I/I2 //

ΩR[S]/A ⊗R[S] B]

[K/K2 // ΩS/A ⊗S B]

(6.4.2)

where the bottom map is a quasi-isomorphism by the known case and the compo-sition is the map that we want to show is a quasi-isomorphism, thus we reduce toshowing that the top map is a quasi-isomorphism. This requires a little trick. SinceR → B is surjective, we can choose, for each s ∈ S, a lift rs ∈ R of f(xs) ∈ B toR. Then the R-algebra map R[S] → R taking xs to rs is surjective and sits in adiagram

A→ R→ R[S]→ R→ B

where the composition R → R is the identity, and the composition R[S]→ B is f .From this diagram we obtain a diagram of complexes

[J/J2 //

ΩR/A ⊗R B]

[I/I2 //

ΩR[S]/A ⊗R[S] B]

[J/J2 // ΩR/A ⊗R B]

(6.4.3)

where the composition is the identity and the top map is the same as the top mapof (6.4.2). But the bottom map in (6.4.3) is an isomorphism by the known case,hence so is the top map.

The variant for sheaves of rings is immediate from the statement about ringsbecause being a quasi-isomorphism can be checked on stalks and the stalk of the


indicated map of complexes at a point x is identied with the map of complexes

[Jx/J2x

//

ΩRx/Ax ⊗Rx Bx]

[Kx/K

2x

// ΩSx/Ax ⊗Sx Bx]

associated to the diagram of rings

Ax → Rx → Sx → Bx

because all constructions involved (tensor product, dierentials, and formation ofkernels) commute with stalks.

Theorem 6.4.2. Suppose A→ B is a map of sheaves of rings which can be factoredas A → S → B, where each stalk Ax → Sx is a ltered direct limit of formallysmooth Ax-algebras and S → B is surjective with kernel K. Then there is a quasi-isomorphism

LB/A∼= [K/K2 // ΩS/A ⊗S B]

of complexes of B-modules.

Proof. It is convenient to refer to S → B as g1 : S1 → B and A[B] → B asg2 : S2 → B. Let G1 and G2 be the kernel of G1 and G2 respectively. Then

[K/K2 // ΩS/A ⊗S B] = [G1/G21

// ΩS1/A ⊗S1 B](6.4.4)

LB/A = [G2/G22

// ΩS2/A ⊗S2 B] .

Let T := S1

⨿S2 (thought of as a sheaf of sets) and let R := A[T ] be the free

A-algebra on T . Note that the stalks of R are free algebras over the stalks of A.If we just had rings instead of sheaves of rings, we could complete the diagram ofA-algebras

Rk1 //

k2

S1

g1

S2g2 // B

(6.4.5)

as indicated by a trick similar to the one used in the proof of Lemma 6.4.1: weuse surjectivity of g1 to choose, for each c ∈ S2, some tc ∈ S1 with g1(tc) = g2(c)and surjectivity of g2 to choose, for each s ∈ S1, some ts ∈ S2 with g2(ts) = g2(s).We then dene k1 : R → S1 to be the unique A-algebra map with k1(xs) = s fors ∈ S1 ⊆ T and k1(xc) = tc for c ∈ S2 ⊆ T and we dene k2 : R → S2 to bethe unique A-algebra map with k1(xs) = ts for s ∈ S1 ⊆ T and k1(xc) = c forc ∈ S2 ⊆ T . Let l : R → B be the common composition and let L be its kernel.


Then we would apply Lemma 6.4.1 to the diagrams

A→ R→ S1 → B(6.4.6)A→ R→ S2 → B

to see that both complexes in (6.4.4) are quasi-isomorphic to

[L/L2 // ΩR/A ⊗R B]

which would complete the proof.

For sheaves of rings there does not seem to be any reasonable way to complete(6.4.5) using any free A algebra R. So we resort to something more roundabout. Letk1 : R → S1 be the unique A-algebra map with k1(xs) = s for s ∈ S1 ⊆ T (we willoften write things like for each s ∈ S1" as an abuse of notation for for each localsection s of S1") and k1(xc) = 0 for c ∈ S2 ⊆ T and let k2 : R → S2 be the uniqueA-algebra map with k2(xs) = 0 for s ∈ S1 ⊆ T and k2(xc) = c for c ∈ S2 ⊆ T . Thesquare (6.4.5) won't commute, so let fi := giki (i = 1, 2) be the two ways aroundit and let Fi be the kernel of the surjection fi : R → B. By applying Lemma 6.4.1to the diagrams (6.4.6) we see that the complexes in (6.4.4) are quasi-isomorphic,respectively, to the complexes

[F1/F21

// ΩR/A ⊗R B](6.4.7)

[F2/F22

// ΩR/A ⊗R B],

thus we reduce to showing that the complexes in (6.4.7) are quasi-isomorphic.

Let h : R → B be the unique A-algebra map with h(xs) = g1(s) for s ∈ S1 ⊆ Tand h(xc) = g2(c) for c ∈ S2 ⊆ T . Let H be the kernel of h. Let j1 : R → Rbe the unique A-algebra map with j1(xs) = xs for s ∈ S1 ⊆ T and j1(xc) = 0 forc ∈ S2 ⊆ T . Let j2 : R → R be the unique A-algebra map with j2(xs) = 0 fors ∈ S1 ⊆ T and j2(xc) = xc for c ∈ S2 ⊆ T . These denitions ensure that hji = fifor i = 1, 2. By applying Lemma 6.4.1 to the diagrams

A // Rj1 // R

h // B

A // Rj2 // R

h // B

we see that the complexes in (6.4.7) are both quasi-isomorphic to the complex

[H/H2 // ΩR/A ⊗R B]

and thus the proof is complete.

Corollary 6.4.3. If A → B is a map of sheaves of rings on a space X such thatBx is a ltered direct limit of formally smooth Ax-algebras for each x ∈ X, then thenatural map LB/A → ΩB/A is a quasi-isomorphism.

Proof. Take S = B in Theorem 6.4.2.


Corollary 6.4.4. If A→ B is a surjective map of sheaves of rings with kernel K,then LB/A is quasi-isomorphic to K/K2[1].

Proof. Take S = A in Theorem 6.4.2. Corollary 6.4.5. Let A → B be a nite type (resp. nitely presented) map ofrings. Then H0(LB/A) = ΩB/A is a nitely generated (resp. nitely presented) B-module. If, furthermore, A (hence also B) is noetherian, then H1(LB/A) is a nitetype (equivalently nitely presented) B-module.

Proof. The rst statement was already proved in Corollary 3.7.4, so we need onlyaddress the second statement. By denition of nite type, we can factor A→ B asA→ S → B with S a polynomial ring over A in nitely many variables and S → Bsurjective. Then ΩS/A is a nitely generated, free S-module. Since A is noetherian,so is S, hence the kernel I of S → B is a nitely generated S-module, and henceI/I2 = I ⊗S B is a nitely generated B-module. Since S is formally smooth over A,Theorem 6.4.2 implies that LB/A is quasi-isomorphic to

[I/I2 → ΩS/A ⊗S B].

The boundary map I/I2 → ΩS/A⊗SB in this complex is a map of nitely generatedB-modules and B is noetherian, so its kernel H1(LB/A) is nitely generated (as isits cokernel ΩB/A).

6.5. Flat base change. As discussed in 3.6, Kähler dierentials commute withdirect limits". This is not true for the cotangent complex, though the cotangentcomplex does commute with at base change" in the following sense:4

Lemma 6.5.1. If (6.3.1) is a pushout (i.e. A′⊗AB → B′ is an isomorphism), thenthe map LB/A ⊗B B′ → LB′/A′ is an isomorphism on H0 and is surjective on H1. IfA→ A′ is also at, then it is a quasi-isomorphism.

Proof. We have a commutative diagram

A //

A[B] //

B

A′ // A′[B] //

B′

A′[B′] // B′

(6.5.1)

where the top two squares are pushouts. Let I (resp. J , K) be the kernel of thesurjection A[B] → B (resp. A′[B] → B′, A′[B′] → B′). Since the upper left squarein (6.5.1) is a pushout, the natural map

ΩA[B]/A ⊗A[B] A′[B] → ΩA′[B]/A′(6.5.2)

4The right point of view is really that the cotangent complex commutes with homotopy directlimits, but it would take us too far aeld to explain this...


is an isomorphism, and since the upper right square is a pushout, the natural map

I ⊗A[B] A′[B] → J(6.5.3)

is surjective and is an isomorphism if A → A′ is at (for then its pushout A[B] →A′[B] is also at). We have a diagram of complexes

[I/I2 ⊗B B′ //

ΩA[B]/A ⊗A[B] B′]

∼=

[J/J2 //

ΩA′[B]/A ⊗A′[B] B′]

[K/K2 // ΩA′[B′]/A ⊗A′[B′] B

′]

where the composition is the natural map LB/A ⊗B B′ → LB′/A′ . On the otherhand, the bottom map is a quasi-isomorphism by Lemma 6.4.1, so it suces toprove that the top map induces an isomorphism on H0 and a surjection on H1 (andan isomorphism when A → A′ is at). The indicated arrow is an isomorphismbecause it is just the isomorphism (6.5.2) tensored over A′[B] with B′. The mapI/I2⊗B B′ → J/J2 is surjective (and an isomorphism when A→ A′ is at) becauseit is just the map (6.5.3) tensored over A′[B] with B′note that

(I ⊗A[B] A′[B])⊗A′[B] B

′ = (I ⊗A[B] B)⊗B B′

= (I/I2)⊗B B′

by commutativity of the upper right square in (6.5.1). The statements about themaps on homology induced by the top map now follow easily.

7. The Truncated Cotangent Complex Continued

In this section we will continue our study of the (truncated) cotangent complexintroduced in 6.

7.1. Transitivity triangle. In a diagram of sheaves of rings A→ B → C, one hasan exact sequence relating the homology of the three cotangent complexes involvedwhich can be viewed as a simultaneous generalization of the exact sequences ofCorollary 3.7.2 and Corollary 3.7.3 in 3.7.

Theorem 7.1.1. If A → B → C are maps of sheaves of rings, there is a naturalexact sequence of C-modules

H1(LB/A ⊗B C) // H1(LC/A) // H1(LC/B) //

// H0(LB/A)⊗B C) // H0(LC/A) // H0(LC/B) // 0

and for any C-module M there is a natural exact sequence of C-modules

0 // Hom(LC/B,M) // Hom(LC/A,M) // Hom(LB/A ⊗B C,M) //

// Ext1(LC/B,M) // Ext1(LC/A,M) // Ext1(LB/A ⊗B C,M)


Proof. Set R := A[B], S := A[C], T := R[C], U := B[C], so we have a commutativediagram of sheaves of rings with cocartesian squares as below.

A //

A[C]

A[B]

xb 7→b

// A[B][C]

xc 7→c

A //

==||||||||B // B[C] // C

=:

A //

S

H

R

I

// T

I

J

@@@

@@@@

@

A //

??~~~~~~~~B // U

K// C

(7.1.1)

To ease notation, we have set R := A[B], S := A[B][C], T := B[C] and we havelabelled each surjection in the right diagram with its kernel. Since the bottom squareis pushout we have

ΩT/R ⊗T U = ΩU/B(7.1.2)

(Formula (3.6.3), 3.6) and since R→ T is at, we also have I = I ⊗R T .

I claim that the rows in the diagram

I/I2 ⊗B C

// J/J2

// K/K2 //

0

0 // ΩR/A ⊗R C // ΩT/A ⊗T C // ΩU/B ⊗U C // 0

(7.1.3)

are exact. From (7.1.2), we have ΩU/B ⊗U C = ΩT/R ⊗T C and we see that thebottom row in (7.1.3) is (7.1.4)⊗TC, where

0→ ΩR/A ⊗R T → ΩT/A → ΩT/R → 0(7.1.4)

is the usual sequence associated to the diagram of rings A→ R→ T , which is exactwith split exact stalks by Proposition 5.3.5 since R→ T has formally smooth stalks,hence it stays exact after tensoring over T with C.

To prove that the top row of (7.1.3) is exact, we rst note that the sequence

0→ I/(I ∩ J2)→ J/J2 → K/K2 → 0(7.1.5)

is exact (this is just a general fact about a composition of surjective ring maps andwas established during the proof of Lemma 6.4.1). We next note that

I/I2 ⊗B C = (I ⊗R B)⊗B C

= (I ⊗R T )⊗T C

= I ⊗T C

= I/(IJ)

so the top row of (7.1.3) is just the natural sequence

I/(IJ)→ J/J2 → K/K2 → 0,

which is exact because (7.1.5) is exact and the natural map I/(IJ)→ I/(I ∩ J2) issurjective since IJ ⊆ (I ∩ J2).


View the columns of (7.1.3) as complexes of C-modules. The right column isLC/B and the left column is LB/A ⊗B C. The middle column is not exactly LC/A,but I claim its homology is naturally isomorphic to that of LC/A. Indeed, we have anatural map of complexes

LC/A = [H/H2 //

ΩS/A ⊗S C]

[J/J2 // ΩT/A ⊗T C]

(7.1.6)

associated to the diagramA→ S → T → C

and this map is a quasi-isomorphism by Lemma 6.4.1 because the stalks of both Sand T are formally smooth algebras over the stalks of A. The map (7.1.6) is hencea quasi-isomorphism from LC/A to the middle column of (7.1.3). The desired exactsequences are hence obtained by applying Lemma 13.8.4 to the diagram (7.1.3). Remark 21. The long exact sequences of Theorem 7.1.1 are natural in the followingsense: Given a diagram of (sheaves of) rings

A //

B //

C

A′ // B′ // C ′

(7.1.7)

the maps induced on homology by the natural maps

LC/B → LC′/B′

LB/A → LB′/A′

LC/A → LC′/A′

determine a C-linear map from the rst exact sequence of Theorem 7.1.1 for the toprow of (7.1.7) to the analogous sequence for the bottom row of (7.1.7). For a C ′-module M , similar remarks apply for the second exact sequence of Theorem 7.1.1.This naturality is immediate from the naturality (in A→ B → C) of 1) the diagram(7.1.1) in the above proof, and 2) the quasi-isomorphism (7.1.6).

Proposition 7.1.2. Suppose A→ B → C is a diagram of sheaves of rings such thateach stalk Bx is a ltered direct limit of formally smooth Ax-algebras and B → C issurjective with kernel I. Then the rst exact sequence of Theorem 7.1.1 coincideswith the one obtained from the Snake Lemma in the exact diagram

0 // 0 //

I/I2

Id // I/I2 //

0

0 // ΩB/A ⊗B CId // ΩB/A ⊗B C // 0 // 0

(7.1.8)

and for a C-module M , the second exact sequence of Theorem 7.1.1 coincides withthe long exact sequence of Ext groups obtained by viewing (7.1.8) as a short exact


sequence of complexes given by the columns and taking Hom into M . In particular,the map

H1(LB/A ⊗B C) → H1(LC/A)

is injective and the connecting map

HomC(LB/A ⊗B C,M) → Ext1C(LC/B,M)

is identied with the map

HomC(ΩB/A ⊗B C,M) → HomB(I/I2,M)

induced by I/I2 → ΩB/A ⊗B C.

Since this proposition is not really logically necessary in anything that follows, wewill leave the proof to the reader.

7.2. The Fundamental Theorem. Here we give two proofs of the FundamentalTheorem of the Cotangent Complex :

Theorem 7.2.1. For a map of rings (or sheaves of rings) A→ B and a B-moduleM , there is a natural isomorphism of B-modules

ExalA(B,M) = Ext1B(LB/A,M).

Denition 18. The image of an algebra extension B ∈ ExalA(B,M) under theisomorphism of Theorem 7.2.1 is called the Kodaira-Spencer class of the extension B,denoted KS(B) ∈ Ext1B(LB/A,M). (C.f. the seminal paper of Kodaira and Spencer[KS].)

First Proof. In this proof we will use only the transitivity triangle (Theorem 7.1.1),the similar-looking exact sequence of Theorem 3.7.1, and the fact that we can provethis theorem in two special cases simply by assembling some of our old results:

Lemma 7.2.2. Theorem 7.2.1 holds in the following special cases:

(1) A→ B is surjective(2) A→ B is formally smooth.

Proof. If A → B is surjective with kernel I, then we saw in Corollary 6.4.4 thatLB/A = I/I2[1]. That is H1(LB/A) = I/I2 and H0(LB/A) = ΩB/A = 0, so

Ext1B(LB/A,M) = HomB(I/I2,M)

by Lemma 13.8.3. But we already saw in Theorem 3.3.1 that there is a naturalisomorphism of B-modules

ExalA(B,M) = HomB(I/I2,M).

Next suppose A → B is a formally smooth map of rings. Then LB/A is quasi-isomorphic to ΩB/A (Corollary 6.4.3) so

Ext1B(LB/A,M) = Ext1B(ΩB/A,M)(7.2.1)


by Lemma 13.8.3. But ΩB/A is a projectiveB-module (Corollary 5.3.3), so Ext1B(ΩB/A,M) =0. Since ExalA(B,M) is also zero (Lemma 5.1.2), the result holds.

For a formally smooth map of sheaves of rings, the natural map LB/A → ΩB/A isstill a quasi-isomorphism, so we still have (7.2.1), but ΩB/A is only locally projective,so Ext1B(ΩB/A,M) may not be zero. But ΩB/A is locally projective, so the highersheaf Exts E xt>0

B (ΩB/A,M) vanish and the local-to-global spectral sequence for Extdegenerates to yield

Ext1B(ΩB/A,M) = H1(X,H omB(ΩB/A,M)).

On the other hand, we can also establish a natural isomorphism

ExalA(B,M) = H1(X,H omB(ΩB/A,M))

as follows: We can compute H1 using ech cohomology,5 so we can view an elementof H1(X,H omB(ΩB/A,M)) as an equivalence class of pairs consisting of an opencover Ui of X and maps sij : ΩB/A|Uij

→ M |Uijwhich satisfy the usual cocycle

condition on the triple intersections Uijk. (The equivalence relation on such pairsis the one generated by saying that such a pair is equivalent to the pair obtainedby passing to any renement of Ui.) The maps sij can be viewed as A-linearderivations Dij : B|Uij

→M |Uij. From these derivations, we can make an A-algebra

extension of B by M by starting with the trivial extension B[M ] on each Ui, thenidentifying these trivial extensions on overlaps using the maps Dij (recall that theautomorphism group of the trivial A-algebra extension B[M ] is the group of A-linearderivations B →M). This yields a map

H1(X,H omB(ΩB/A,M)) → ExalA(B,M).

This map is easily seen to be an isomorphism because formal smoothness of A→ Bimplies that every A-algebra extension B′ ∈ ExalA(B,M) is locally trivial, hence isof the form given by the construction.

Now suppose A→ B is an arbitrary map of rings. We can factor it as A→ S → Bwhere A→ S is formally smooth and S → B is surjective with kernel I. To be surethat everything is functorial, we can take the usual factorization with S = A[B] (itdoesn't really matter...). We next compare the exact sequence of Theorem 3.7.1 andthe one from Theorem 7.1.1 to get a commutative diagram with exact rows

DerA(S,M)δ // ExalS(B,M) // ExalA(B,M) // ExalA(S,M)

Hom(LS/A ⊗S B,M)

∼=

OO

// Ext1(LB/S,M)

∼=

OO

// Ext1(LB/A,M)

OO

// Ext1(LS/A,M)

∼=

OO(7.2.2)

5Although ech cohomology does not generally agree with the usual derived functor cohomology,they always agree on H1.


where the left vertical arrow is the sequence of natural isomorphisms

Hom(LS/A ⊗S B,M) = HomB(H0(LS/A ⊗S B),M)

= HomB(H0(LS/A)⊗S B,M)

= HomB(ΩS/A ⊗S B,M)

= HomS(ΩS/A,M)

= DerA(S,M)

(the only nontrivial isomorphism here is the fact that LS/A → ΩS/A is a quasi-isomorphism by Corollary 6.4.3) and the next vertical arrow is the rst case ofLemma 7.2.2. One sees that the square commutes using the explicit description ofthe map δ in Theorem 3.7.1 and the map

Hom(LS/A ⊗S B,M) → Ext1(LB/S,M)

in Proposition 7.1.2. But we already saw in the course of proving the second caseof Lemma 7.2.2 that the terms on the right of both exact sequences are zero, so weget an isomorphism given by the dotted arrow as indicated.

For sheaves of rings, we can at least choose our factorization A → S → B sothat the stalks of S are formally smooth over the stalks of A, in which case we stillhave the exact diagram (7.2.2), but now the terms on the right needn't be zero,though we did establish an isomorphism between them as indicated (the secondcase of Lemma 7.2.2). In this case we run into a snag because we haven't actuallyconstructed the dotted arrow making the diagram commute, but if we did have suchan arrow, it would automatically be an isomorphism by the Slightly Subtle FiveLemma. That is, we could complete the proof if we just had some natural mapfrom ExalA(B,M) to Ext1(LB/A,M) (or in the other direction) which reduced tothe isomorphisms of Lemma 7.2.2 in those cases. We will construct natural maps inboth directions in the Second Proof, so we will be content to leave the rst proof abit incomplete for the case of sheaves of rings.

Second Proof. The second proof uses almost nothing but the denitions. We will usethe Yoneda description of Ext1 (end of 13.8). We will begin by proving a variantof the Fundamental Theorem which illustrates all the key ideas, though we will notactually need to make any use of this variant.

Suppose A → B is a map of sheaves of rings, M is a B-module, and E ∈Ext1B(ΩB/A,M). Using the Yoneda description of Ext1, we can view E as (an iso-morphism class of) a short exact sequence of B-modules

0 // M // Eg // ΩB/A

// 0.(7.2.3)

Pulling this sequence back along the universal A-linear derivation d : B → ΩB/A weobtain an exact sequence

0 // M // B′ // B // 0(7.2.4)


where B′ := B ×ΩB/AE and B′ → B is given by (b, e) 7→ b.6

A priori (7.2.4) is just an exact sequence of A-modules. However, we can denea multiplication on B′ by the rule

(b1, e1)(b2, e2) := (b1b2, b1e2 + b2e1).

The right hand side is a priori just a local section of B ×E; we need to check thatit is actually in B ×ΩB/A

E. To see this, we compute

d(b1b2) = b1db2 + b2db1

= b1g(e2) + b2g(e1)

= g(b1e2 + b2e1)

using the Leibnitz rule and the fact that dbi = g(ei) (i = 1, 2). Now that we knowthis multiplication is well-dened it is trivial to see that B′ is a sheaf of commutativerings, the map A → B′ given by a 7→ (a, 0) (notation for A → B suppressed) is amap of sheaves of rings, and B′ → B is a surjective map of A-algebras with kernelM = (0,m) ∈ B′ : m ∈ M. In other words, we have shown that the inclusionB′ ⊆ B[E] = B×E of B′ into the trivial square zero extension of B by E is actuallyan inclusion of A-algebras over B even though a priori it is only an inclusion ofA-modules over B.

The addition and scalar multiplication on the B-module Ext1B(ΩB/A,M) coincide,via the Yoneda description of Ext1, with the Baer sum and scalar multiplication ofextensions of the form (7.2.3) much like the one we dened on ExalA(B,M) in 3.3.These operations are compatible with the pullback construction E 7→ B′ above, soour construction E 7→ B′ denes a map of B-modules

Ext1B(ΩB/A,M) → ExalA(B,M).(7.2.5)

Proposition 7.2.3. The map (7.2.5) denes an injective map of B-modules whoseimage in ExalA(B,M) is the set ExalA(B,M) of A-algebra extensions of B by M

0 // M // B′ // B // 0

A

`ÀAAAAAAA

OO

such that the sequence

0→M → ΩB′/A ⊗B′ B → ΩB/A → 0(7.2.6)

is exact (by Corollary 3.7.3 the only issue is the injectivity of the left map m 7→dm⊗ 1). The inverse of

Ext1B(ΩB/A,M) → ExalA(B,M)(7.2.7)

6The sheaf of sets underlying an inverse limit of modules coincides with the inverse limit of theunderlying sheaves of sets. Inverse limits of sheaves of sets coincide with inverse limits taken inpresheavesthey are taken objectwise" (i.e. on each open set).


is the Kodaira-Spencer map

ExalA(B,M) → Ext1B(ΩB/A,M)(7.2.8)

taking B′ ∈ ExalA(B,M) to the extension (7.2.6).

Proof. Consider the algebra extension B′ = B ×ΩB/AE constructed from E ∈

Ext1B(ΩB/A,M) as above. We have a solid diagram with exact rows

Mδ // ΩB′/A ⊗B′ B

// ΩB/A// 0

0 // M // Eg // ΩB/A

// 0

(the top row is exact by Corollary 3.7.3). If we can construct the dotted arrow asindicated making the diagram commute, the map δ must be injective, and then thedotted arrow must be an isomorphism by the Five Lemma. Note ΩB′/A ⊗B′ B =ΩB′/A/MΩB′/A. It is straightforward to check that d(b, e) 7→ e is well-dened andwill serve as the desired dotted arrow. This proves that (7.2.5) actually denes amap (7.2.7) and that the composition

(7.2.8) (7.2.7) : Ext1B(ΩB/A,M) → Ext1B(ΩB/A,M)

is the identity. To see that the other composition

(7.2.7) (7.2.8) : ExalA(B,M) → ExalA(B,M)

is the identity, it is enough to construct, given B′ ∈ ExalA(B,M), a map of A-algebraextensions of B by M

0 // M // B′

h // B // 0

0 // M // B′′ // B // 0

when B′′ = B ×ΩB/AΩB′/A/MΩB′/A (such a map is necessarily an isomorphism).

The map b′ 7→ (h(b′), db′) will do the job.

The second proof of Theorem 7.2.1 is just a jazzed up version of the above argu-ment (it is in many ways even easier). Let A → B be a map of sheaves of rings,M a B-module. Let S = A[B], and let A → S → B be the usual factorization ofA → B into a map with formally smooth stalks followed by a surjection. Let I bethe kernel of S → B, so

LB/A = [I/I2 → ΩS/A/IΩS/A].

(In reality any such factorization would be good enough for what follows but it makesthings a little simpler and a little more obviously functorial to use the functorialfactorization.)

Given an A-algebra extension

0→M → B′ → B → 0(7.2.9)


of B by M , we pull back along S → B to get an A-algebra extension

0→M → S ′ → S → 0(7.2.10)

of S by M (regarded as an S-module via S → B) with S ′ := S ×B B′. The kernelof S ′ → B′ is given by (i, 0) ∈ S ′ : i ∈ I. There is no harm in abusively callingthis ideal I as it is mapped isomorphically onto I ⊆ S under S ′ → S. Since A→ Shas formally smooth stalks, the sequence

0→M → ΩS′/A ⊗S′ S → S → 0(7.2.11)

is exact and has split exact stalks (Proposition 5.3.4), hence it remains exact aftertensoring over S with B, so we obtain a diagram

I/I2

Id // I/I2

0 // M // ΩS′/A ⊗S′ B // ΩS/A/IΩS/A

// 0

(7.2.12)

where the bottom row is exact. As usual, we view it as a short exact sequence ofcomplexes given by the columns. The left column is M (viewed as a complex sittingin degree 0) and the right column is LB/A, so (7.2.12) is in Ext1B(LB/A,M). Thisdenes a map

ExalA(B,M) → Ext1B(LB/A,M),(7.2.13)

called the Kodaira-Spencer map. One could check, with a certain amount of tedium,that (7.2.13) is B-linear, but this won't be necessary because we will now constructan inverse to (7.2.13) which is more easily seen to be B-linear.

An element of Ext1B(LB/A,M) can be viewed as an extension of complexes of LB/A

by M , which necessarily takes the form

I/I2

f

Id // I/I2

0 // M // E

g // ΩS/A/IΩS/A// 0

(7.2.14)

where the row is exact (see the end of 13.8). Using just the bottom row of (7.2.14),we can form an A-algebra extension

0→M → S ′ → S → 0

of S by M by setting S ′ := S ×ΩS/A/IΩS/AE. The implicit map S → ΩS/A/IΩS/A

is of course the universal A-linear derivation d : S → ΩS/A followed by the quotientmap ΩS/A → ΩS/A/IΩS/A and the multiplication for the ring structure on S ′ is givenby

(s1, e1)(s2, e2) = (s1s2, s1e1 + s2e2).

The map f in (7.2.14) yields a map I → S ′ given by i 7→ (i, f(i)), which is well-dened by commutativity of the square in (7.2.14). This map is clearly injectiveand it is straightforward to check that its image is an ideal I ⊆ S ′. This ideal is


clearly taken bijectively onto I ⊆ S by S ′ → S. Taking quotients of S and S ′ by Iwe obtain a diagram with exact rows

0 // I // S ′

// B′

// 0

0 // I // S // B // 0

(7.2.15)

where B′ := S ′/I. The Snake Lemma implies that B′ → B is surjective with kernelM , hence we have an A-algebra extension

0→M → B′ → B → 0(7.2.16)

of B by M . This construction denes a map

Ext1B(LB/A,M) → ExalA(B,M).(7.2.17)

It is straightforward to check that (7.2.17) is B-linear because the whole construc-tion is phrased in terms of extensions (the addition for Ext1B(LB/A,M) takes twoextensions E1, E2 of the form (7.2.14) to another complex E1+E2 of this form wherethe bottom row of E1 +E2 is the usual Baer sum of the bottom rows of E1, E2 andthe map f for E1 + E2 is f1 × f2...).

It remains to prove that (7.2.13) and (7.2.17) are inverse. This is not appreciablydierent from what we did in the proof of Proposition 7.2.3, one needs only to keeptrack of the extra map out of I/I2. The second proof is complete.

7.3. Curvilinear algebra extensions. This section is essentially an extended ex-ample intended to familiarize the reader with the B-module ExalA(B,M) and theFundamental Theorem of the Cotangent Complex in 7.2.

Proposition 7.3.1. For any ring A and any n ∈ 2, 3, . . . , let B be the A-algebraA[x]/(xn). There is a natural isomorphism of B-modules

ExalA(B,A) = A,

where A is regarded as a B-module here via the unique A-algebra map B → Akilling x. Indeed, ExalA(B,A) is freely generated as an A-module by the A-algebraextension

0 // Axn

// A[x]/(xn+1)x 7→x // B // 0.

Proof. The B-module A is annihilated by x, hence the B-module ExalA(B,A) isalso annihilated by x (Remark 10), so ExalA(B,A) is a B/(x) = A module regardedas a B-module by restriction of scalars along B → A. The nal statement of theproposition is tantamount to saying that for any B′ ∈ ExalA(B,A), there is a uniqueelement a ∈ A for which there is a morphism of A-algebra extensions as below.

0 // A

a·_

xn// A[x]/(xn+1)

x7→x //

B // 0

0 // A // B′g // B // 0

(7.3.1)


Given B′ ∈ ExalA(B,A) as in the bottom row above, choose some y ∈ B′ withg(y) = x. Then g(yn) = g(y)n = 0, so a := yn is in A ⊆ B′. We claim yn+1 = 0in B′. Indeed, yn+1 = yyn = x · a = 0 by denition of the B-module structureon the square zero ideal A and the fact that x ∈ B annihilates the B-moduleA. We can then obtain a map of A-algebra completing (7.3.1) as indicated bymapping x ∈ A[x]/(xn+1) to y ∈ B′. For uniqueness, suppose we have another suchcompletion using another a′ ∈ A. Let y′ ∈ B′ be the image of x ∈ A[x]/(xn+1)under this other completion. Then by commutativity of the square on the right,y′ ∈ B′ is also a lift of x and by commutativity of the square on the left we musthave a′ = (y′)n. Since y, y′ ∈ B′ both lift x ∈ B we can write y′ = y + i for somei ∈ A. Then we compute

a′ = (y′)n = (y + i)n = yn + nyn−1i = yn = a

because A is square zero in B′ (so the terms in the binomial expansion of (y + i)n

involving higher powers of i are zero in B′) and yn−1i = xn−1 · i = 0 because of theway A is regarded as a B-module via the algebra extension B′ and the fact thatx ∈ B annihilates A (notice that this is the point at which we use n > 1).

On the other hand, we know from the Fundamental Theorem (Theorem 7.2.1)that

ExalA(B,A) = Ext1B(LB/A, A)

in the situation of Proposition 7.3.1 so we ought to be able to show directly that

Ext1B(LB/A, A) = A.

The cotangent complex LB/A in this situation is easily understood (up to quasi-isomorphism): Since A[x] is a free (hence formally smooth) A-algebra and we havean obvious A-algebra surjection A[x]→ B = A[x]/(xn) with kernel I = (xn) (a freeA[x]-module of rank one), we see from Theorem 6.4.2 that LB/A is quasi-isomorphicto the complex

[(xn)/(x2n)δ // ΩA[x]/A ⊗A[x] B] ∼= [B

nxn−1// B](7.3.2)

Let us assume for a moment that n = 0 in A (equivalently in B). Then LB/A isquasi-isomorphic to the cohomologically formal" complex B ⊕B[1] consisting of Bin degree zeros 1, 0, with boundary map zero, so we nd

Ext1B(LB/A, A) = Ext1B(B,A)⊕ HomB(B,A)

= A

using Lemma 13.8.2 for the rst equality and the fact that Ext1B(B,A) = 0 (becauseB is free, hence projective) for the second equality.

In general it is not so easy to show that Ext1B(LB/A, A) = A. This follows fromthe description of the quasi-isomorphism type of LB/A in (7.3.2) together with thefollowing general lemma whose proof we will sketch in Exercise 12:


Lemma 7.3.2. Let B be a ring, b ∈ B, L = [_ · b : B → B] the Koszul complex"of b (a two term complex of B-modules in degrees 1, 0). Let B → A be a ringhomomorphism killing b. Then Ext1B(L,A) = A.

7.4. The case of schemes. In this section, we make a few remarks about thecotangent complex of a morphism of schemes f : X → Y , which is, by denition,the cotangent complex of f as a morphism of (locally) ringed spaces, as dened inDenition 17.

Theorem 7.4.1. Let f : A → B be a map of rings, X := SpecB, Y := SpecA,g : X → Y the map induced by f . Then the complex of OX-modules LX/Y isnaturally quasi-isomorphic to the complex of quasi-coherent OX-modules L∼B/A.

Proof. The proof is basically the same as the proof of Theorem 3.9.1. We have apushout diagram

B // OX

A

f

OO

// g−1OY

g♯

OO

of sheaves of rings on X where the horizontal arrows are at (their stalks are local-izations), hence the natural map

LB/A ⊗B OX → LX/Y

is a quasi-isomorphism by Lemma 6.5.1. Since formation of the cotangent complexcommutes with change of space" (6.3), we have LB/A = LB/A and we have

LB/A ⊗B OX = L∼B/A,

whence the result. Corollary 7.4.2. Let f : X → Y be a map of schemes. Then the cotangentcomplex LX/Y is locally quasi-isomorphic to a 2-term complex of quasi-coherentsheaves [L1 → L0]. In particular, the cohomology sheaves of LX/Y are quasi-coherentOX-modules. If f is a locally nite type morphism of locally noetherian schemes,then the cohomology sheaves of LX/Y are coherent. If f is formally smooth, thenH1(LX/Y ) = 0.

Proof. The questions are local, so Theorem 7.4.1 immediately yields the rst state-ment and reduces the niteness statement to Corollary 6.4.5 and the nal statementto Corollary 6.4.3 (applied just in the case of a map of rings as opposed to sheavesof rings).

7.5. Smoothness and the cotangent complex. The purpose of this section isto characterize smoothness and formal smoothness of a map of schemes f : X → Yin terms of properties of the cotangent complex LX/Y .

Theorem 7.5.1. Let f : X → Y be a map of schemes. The following are equivalent:


(1) f is formally smooth (Denition 10).(2) For every open subspace U ⊆ X and every quasi-coherent sheaf G on U ,

every (f−1OY )|U-algebra extension of OU by G is locally trivial.(3) For every ane open subspace U ⊆ X and every quasi-coherent sheaf G on

U , every (f−1OY )|U-algebra extension of OU by G is trivial.(4) For every open subspace U ⊆ X and every quasi-coherent sheaf G on U ,

E xt1(LU/Y ,G ) = 0.(5) H1(LX/Y ) = 0 and H0(LX/Y ) = ΩX/Y is Qco locally projective (Deni-

tion 27).

If f is of locally nite presentation, these conditions are also equivalent to:

(3) f is smooth.(4) H>0(LX/Y ) = 0 and H0(LX/Y ) = ΩX/Y is locally free of locally nite rank.

If f is of locally nite presentation and X is locally noetherian, these conditions arealso equivalent to:

(5) E xt1(LX/Y ,G ) = 0 for every coherent sheaf G on X.

Proof. For (1) implies (2), rst note that if f is formally smooth, then so is f |U :U → Y (Lemma 5.5.1), so we can reduce to establishing the statement in (2) whenU = X. Let

0→ G → OX′ → OX → 0(7.5.1)

be an f−1OY -algebra extension of OX by G . Then X ′ := (X,OX) is a scheme(Corollary 2.2.3) ane whenever X is ane, and we have a commutative diagramof schemes

X //

X

f

X ′

>>

// Y

when the left map is the square-zero thickening X → X ′ corresponding to thesurjection OX′ → OX . Since f is formally smooth, there is locally a lift as indicated(because X is locally ane); such a lift is the same thing as an f−1OY -algebrasection of OX′ → OX (a local trivialization of (7.5.1)).

For (2) implies (3), use Lemma 3.5.1 to see that the obstruction to gluing localtrivializations (which exist by hypothesis) of such an algebra extension into a globaltrivialization lies in H1(U,H om(ΩU/Y ,G )), which vanishes when U is ane becauseH om(ΩU/Y ,G ) is quasi-coherent (Corollary 13.5.7).

Obviously (3) implies (2) since we can cover any U with open anes. We haveproved that (2) and (3) are equivalent.

For (3) implies (1), we rst reduce to the case where X = SpecB and Y = SpecAare ane using Lemma 5.5.1 (it helps here that we already know (2) and (3) areequivalent). The hypothesis in (3) says that ExalY (X,G ) = 0 for each quasi-coherent


sheaf on X, which is equivalent to saying that ExalA(B,M) = 0 for each B-moduleM (c.f. Proposition 2.2.1). This latter vanishing property is equivalent to formalsmoothness of the ring map A → B (Lemma 5.1.2), which is equivalent to formalsmoothness of f (Remark 15).

For (2) implies (4), we rst note that E xt1U(LU/Y ,G ) is the sheaf associated tothe presheaf

V 7→ Ext1V (LU/Y |V,G |V ) = Ext1V (LV/Y ,G |V )

on U (this is a variant of Lemma 13.5.2 for complexes as mentioned in Remark 33).We want to prove that any section s of E xt1U(LU/Y ,G ) over any open subset V ⊆U is zero. Such an s is represented by a cover Vi of V and elements si ∈Ext1Vi

(LVi/Y ,G |Vi) satisfying a certain compatibility on the overlaps Vij. We canprove that s = 0 locally, so it is enough to prove that each si restricts to zero on acover of Vi. By the Fundamental Theorem 7.2.1, we have

ExalY (Vi,G |Vi) = Ext1(LVi/Y ,G |Vi)

so the hypothesis in (2) that each element of the left hand side is locally trivialimplies that each si is locally zero as desired.

For (4) implies (5), we make use of the exact sequence of OU -modules

0 // E xt1(ΩU/Y ,G ) // E xt1(LU/Y ,G )

// H om(H1(LU/Y ),M) // E xt2(ΩU/Y ,G )

(7.5.2)

(the sheaed version of the exact sequence (13.8.3) of 13.8). (Note thatH0(LU/Y ) =

ΩU/Y .) From (7.5.2) and the hypothesized vanishing of E xt1(LU/Y ,G ) in (4) weconclude that E xt1U(ΩU/Y ,G ) = 0 for every open U ⊆ X and every quasi-coherentsheaf G on U , so ΩX/Y is Qco locally projective in the sense of Denition 27,which proves the rst part of (5). Since we know H1(LX/Y ) is quasi-coherent, wecan prove that H1(LX/Y ) = 0 by proving that H omX(H1(LX/Y ),G ) = 0 for ev-ery quasi-coherent sheaf G on X. This in turn would follow from the hypothesizedvanishing of E xt1(LX/Y ,G ) and the exactness of (7.5.2) (with U = X) providedwe could prove that the map E xt1(LX/Y ,G ) → H om(H1(LX/Y ,G ) appearing in(7.5.2) is surjective; this in turn is equivalent to saying that the map

H om(H1(LX/Y ),G ) → E xt2(ΩX/Y ,G )(7.5.3)

appearing in (7.5.2) is the zero map. This would be obvious if we knew thatE xt2(ΩX/Y ,G ) = 0. On a locally noetherian scheme, this vanishing is obviousbecause the comparison map

E xt2Qco(X)(ΩX/Y ,G ) → E xt2X(ΩX/Y ,G )(7.5.4)

is an isomorphism (Remark 36) and the left hand side vanishes (without noetherianhypotheses) because ΩX/Y is Qco locally projective (Theorem 13.6.5). The proofcan be completed in the general case by proving the following Claim: (7.5.3) factors


(at least locally) through the map (7.5.4). The Claim follows from the followingobservations:

(1) LX/Y is, at least locally, quasi-isomorphic to a 2-term complex L = [L1 → L0]of quasi-coherent sheaves (Corollary 7.4.2).

(2) If L → L′ is a quasi-isomorphism, then the exact sequences analogous to(7.5.2) (but with LX/Y replaced with L or L′) are isomorphic (this is obvious).

(3) If L = [L1 → L0] is a 2-term complex of quasi-coherent sheaves, then theconnecting map"

H om(H1(L),G )→ E xt2(H0(L),G )

appearing in the sequence analogous to (7.5.2) for L factors through thecomparison map"

E xt2Qco(X)(H0(L),G )→ E xt2(H0(L),G )

(the map (13.6.5) discussed in 13.6). (This follows from the explicit descrip-tion of this connecting map given in Remark 39.)

For (5) implies (2): It is clear that (2) is local in nature and that the proper-ties of LX/Y in (5) are inhertied by open subschemes of X, so we can reduce tothe case where X = SpecB and Y = SpecA are ane. It suces to show thatExalY (X,G ) = 0 for each quasi-coherent sheaf G on X. This is equivalent to say-ing that ExalA(B,M) = 0 for each B-module M (c.f. Proposition 2.2.1). SinceΩX/Y = Ω∼B/A and H1(LX/Y ) = H1(LB/A)

∼ (Theorem 7.4.1), the hypotheses in (5)implies that H1(LB/A) = 0 and that H0(LB/A) = ΩB/A is projective. Then wecompute

ExalA(B,M) = Ext1(LB/A,M)

= Ext1(ΩB/A,M)

= 0

using the Fundamental Theorem 7.2.1, Lemma 13.8.3, and the projectivity of ΩB/A.

7.6. Representable obstruction theory. Fix a base scheme S and an S-schemeY (a map of schemes Y → S). We think of the category Sch/Y of schemes overY as a moduli problem (category bered in groupoids) over the category Sch/Sof schemes as in 1.2. This moduli problem is (1.2.1) (with the addition of thebase scheme S). We will now construct an Aut/Def/Ob theory, in the sense of1.6, called the canonical representable obstruction theory for this moduli problem.Everything we are about to do would make sense with schemes" replaced by ringedspaces," locally ringed spaces," etc., but for the sake of concreteness let us just sayschemes."

An object of our moduli problem" over an S-scheme X is a map of S-schemesf : X → Y . There are no automorphisms" of objects for this moduli problem (theforgetful functor Sch/Y → Sch/S is faithful). Given a map of S-schemes X → X ′


and such an f , a lifting of f to X ′" is a completion as indicated in the diagram ofS-schemes below.

X //

f

X ′

f ′~~Y

(7.6.1)

We are interested in the question of existence of such completions (and the classi-cation of such completions) when X → X ′ is a square zero thickening of S-schemes(2.2). Given an S-schemeX, anOX-moduleM and a map of S-schemes f : X → Y ,we set

Aut(X,M, f) := 0

Def(X,M, f) := HomX(f∗ΩY/S,M)

Ob(X,M, f) := Ext1f−1OY(f−1LY/S,M).

Proposition 7.6.1. For any scheme Y , there is an Aut/Def/Ob theory for Sch/Y(1.6) with the automorphism, deformation, and obstruction groups above. In otherwords, if X → X ′ is a square zero closed embedding of S-schemes with ideal M andf : X → Y is a map of S-schemes, then there is an element

ob(X → X ′, f) ∈ Ext1f−1OY(f−1LY/S,M)

whose vanishing is necessary and sucient for the existence of a completion in(7.6.1). When this obstruction vanishes, the set of completions in (7.6.1) is a torsorunder HomX(f

∗ΩY/S,M).

Proof. The square-zero thickening X → X ′ is an isomorphism on topological spaces(Lemma 2.1.1) and a completion in the diagram of S-schemes (7.6.1) is the samething as a completion

0 // M // OX′ // OX// 0

f−1OY

OOdd(7.6.2)

of g−1OS-algebras, where g : X → S is the structure map for the S-scheme X (andalso the structure map for the S-scheme X ′ on the level of spaces, suppressing theisomorphism of spaces X ∼= X ′). If we pull back the g−1OS-algebra extension in(7.6.2) along f−1OY → OX , we obtain a g−1OS-algebra extension

0 // M // E // f−1OY// 0

g−1OS

OObbEEEEEEEEE

(7.6.3)

and a completion in (7.6.2) is the same thing as a trivialization of the g−1OS-algebraextension (7.6.3). By invariance under change of space (6.3), the cotangent complexof g−1OS → f−1OY is f−1LY/S. The obstruction to nding such a trivialization is


the image ob(X → X ′, f) of the algebra extension (7.6.3) in Ob(X,M, f) under theisomorphism

Exalg−1OS(f−1OY ,M) = Ext1f−1OY

(f−1LY/S,M)

given by the Fundamental Theorem of the Cotangent Complex (Theorem 7.2.1).

Corollary 7.6.2. Let f : X → Y be a locally nitely presented map of schemes.Then f is smooth (i.e. formally smooth) i ΩX/Y is locally free (of locally nite rank)and H1(LX/Y ) = 0.

Proof. It is clear from Proposition 5.5.3 and Corollary 7.4.2 that f smooth impliesthe two conditions. Now assume H1(LX/Y ) = 0 and ΩX/Y is locally free of locallynite rank and let us prove that f is formally smooth. We need to nd a lift in anydiagram

T

i // X

f

T ′ // Y

where T → T ′ is a square-zero thickening of ane schemes with ideal I ∈Mod(OT ).Note that I is quasi-coherent. By the proposition, it suces to prove that

Ext1i−1OX(i−1LX/Y , I) = 0.

We are regarding I as an i−1OX module via restriction of scalars along i−1OX → OT .Since H1(LX/Y ) = 0 and i−1 is exact, we have H1(i

−1LX/Y ) = 0 and H0(i−1LX/Y ) =

i−1H0(LX/Y ) = i−1ΩX/Y , so

Ext1i−1OX(i−1LX/Y , I) = Ext1i−1OX

(i−1ΩX/Y , I)

by Lemma 13.8.3. To show this latter Ext group is zero, we need to show that anyi−1OX-module extension

0→ I → E → i−1ΩX/Y → 0

splits. Since ΩX/Y is a locally free OX-module of locally nite rank, i−1ΩX/Y isalso a locally free i−1OX-module of locally nite rank, so one can certainly nd asplitting locally. The sheaf of such local splittings is a torsor under

H omi−1OX(i−1ΩX/Y , I) = H omOT

(i∗ΩX/Y , I),

so the obstruction to nding a global splitting lies in

H1(T,H omT (i∗ΩX/Y , I)),

but this vanishes because H omT (i∗ΩX/Y , I) is quasi-coherent and T is ane.

7.7. Obstruction theory revisited.


8. The Cotangent Complex

We have devoted the last two sections to the study of an object that we calledthe truncated cotangent complex. The purpose of this section is to survey (withoutproofs) the basic theory of the (full, untruncated) cotangent complex. We will usesome of these results in the later sections.

8.1. Denition of the cotangent complex. We begin with the denition. Sup-pose A → B is a map of rings (or sheaves of rings). In dening the truncatedcotangent complex (6) we started by observing that A → B can be functoriallyfactored as A→ A[B]→ B where A[B] is the polynomial ring" over A in variablesxb (one for each b ∈ B) and A[B] → B is the tautological A-algebra surjectionxb 7→ b. In fact we can proceed further: Consider next the free A-algebra A[A[B]]on the set underlying A[B]. This is a polynomial ring" over A with one variable ycfor each c =

∑b abxb in A[B]. There are now two dierent maps A[A[B]] ⇒ A[B],

namely yc 7→ c and yc 7→ x∑b abb

, the composition of each with A[B] → B is givenby yc 7→

∑b abb. There is also an A-algebra map A[B]→ A[A[B]] given by xb 7→ yxb

which simultaneously sections both maps A[A[B]] ⇒ A[B]. We can keep going: forexample, there will be three dierent obvious" maps from A[A[A[B]]] to A[A[B]],and so forth. If we set P0(B) = A[B], P1(B) = A[P0(B)], P2(B) = A[P1(B)], andso forth, then the Pn(B) can be given the structure of a simplicial A-algebra P (B).

What is going on is this: We have the forgetful functor V (V " for vergesslich"we can't use F" because that has another use) from A-algebras to sets, and its leftadjoint F , the free A-algebra functor. So we have natural transformations FV → Idand Id→ V F . The evaluation of FV → Id on an A-algebra B is the map A[B]→ Bgiven by xb 7→ b discussed above. From the pair of adjoint functors (F, V ) we canproduce [Ill, 1.5] a simplicial endofunctor P on A-algebras by setting

Pn := FV FV · · ·FV︸︷︷︸n+1 copies of FV

.

The boundary maps δi : Pn → Pn−1 (i = 0, . . . , n) are dened by contractingthe ith copy (counting from right to left, starting with 0) of FV " in Pn to Idusing the adjunction FV → Id. The degeneracy maps si : Pn−1 → Pn are denedsimilarly by inserting" an extra V F at the appropriate point using the adjunctionmap Id → V F . Evaluating this simplicial endofunctor P at an A-algebra B givesthe simplicial A-algebra P (B) mentioned above. The adjunction map FV → Idalso gives P an augmentation map P → Id whose evaluation at B is the evidentaugmentation P (B) → B given by xb 7→ b. The map P (B) → B is called thestandard free resolution of B as an A-algebra. It is a resolution in a sense we willmake more precise momentarily; it is free" in the sense that each Pn(B) is a freeA-algebra.

Taking dierentials relative to A and tensoring up to B, we get a simplicial B-module ΩP (B)/A ⊗P (B) B, which we can view as a complex of B-modules

· · · → ΩP1(B)/A ⊗P1(B) B → ΩP0(B)/A ⊗P0(B) B → 0→ · · ·


supported in non-negative degreesthis is, by denition, the cotangent complexLB/A of A → B. (The boundary maps are the alternating sums of the maps onKähler dierentials induced by the boundary maps δi for P , tensored up to B.)Note that each Pn(B) is free over A, so this is a complex of free B-modules. All ofthe same constructions make sense for sheaves of rings, though one only concludes atthis last point that the stalks of the ΩPn(B)/A are free, so at least LB/A is a complexof at B-modules. The augmentation map P (B)→ B gives an augmentation mapLB/A → ΩB/A.

Whenever one constructs a simplicial endofunctor P from adjoint functors (F, V )as above, one can also show [Ill, 1.5.3] that:

(1) The augmention V P → Id is a homotopy equivalence.(2) PF → F is a homotopy equivalence.

(One makes use of the natural transformations FV → Id and Id→ V F to constructthe homotopy inverse and homotopies.)

In our situation, (1) means that for any A-algebra B, the map P (B) → B ofsimplicial A-algebras becomes a homotopy equivalence V P (B) → V (B) on thelevel of underlying simplicial sets (the homotopy inverse and the homotopies areall functorial in B). The augmention map of simplicial sets V P (B) → V (B) alsounderlies a map of simplicial abelian groupsnamely the additive abelian groups forthese A-algebras, so the homotopy groups of V P (B) are just the homology groupsof the chain complex associated to the simplicial abelian group underlying P (B).So P (B)→ B is a resolution in the sense that the complex of abelian groups

· · · → P2 → P1 → P0 → B → 0

is acyclicthe boundary map Pn → Pn−1 here is the alternating sum of the mapsδi : Pn → Pn−1 (this is not a ring homomorphism of course!).

In our situation, (2) means that for any set S, the augmentation map PF (S)→F (S) is a homotopy equivalence of simplicial A-algebras. Of course this stays ahomotopy equivalence after applying any functor, so after taking Kähler dierentials,we see that ΩPF (S)/A⊗PF (S)F (S)→ ΩF (S)/A is a homotopy equivalencethis proves:

Lemma 8.1.1. The augmentation map LB/A → ΩB/A is a quasi-isomorphism forevery free A-algebra B.

It might be helpful to explain a manifestation of this general nonsense which mightbe more familiar to the reader. If X is a topological space, then the product F ofthe stalk functors

F : Ab(X) →∏x∈X

Ab(8.1.1)

F 7→∏x∈X

Fx


is left adjoint to the product of the pushforwards"

V :∏x∈X

Ab → Ab(X)(8.1.2)

(Ax) 7→∏x∈X

x∗Ax.

The same sort of general nonsense construction yields a cosimplicial endofunctor Gon Ab(X) (with G0 = V F , G1 = V FV F , and so forth) with an augmentation Id→G that becomes a homotopy equivalence after applying F . If we evaluate at F ∈Ab(X), we thus obtain a cosimplicial object GF in Ab(X) with an augmentationF → GF . By taking alternating sums of the boundary maps from GF , this givesa complex

0→ F → G0F → G1F → · · ·(8.1.3)

in Ab(X) whose image under F is contractiblein other words, the stalks of (8.1.3)are contractible, so (8.1.3) is a resolution of F . In fact it is a resolution of F byasque sheaves and is the canonical asque resolution of Godement.

It is clear that all the constructions involved in the denition of LB/A commutewith ltered direct limits, so formation of LB/A commutes with change of topologicalspace, and, in particular, with formation of stalks, just as was the case for thetruncated cotangent complex.

Lemma 8.1.2. Let A be a ring (or sheaf of rings), M a at A-module, B :=Sym∗AM . Then the augmentation map LB/A → ΩB/A is a quasi-isomorphism andthere is a natural isomorphism ΩB/A = M ⊗A B.

Proof. We can check all these statements on stalks, so the statements for sheaves ofrings follows from the statement for rings. Using Lazard's Theorem, we write M asa ltered direct limit of nitely generated free A-modules Mi. The result holds foreach Mi by Lemma 8.1.1 since Sym∗A Mi is a free A-algebra, hence the result holdsfor M because everything commutes with ltered direct limits.

We should justify our denition of the truncated cotangent complex." Illusieshows [Ill, III.1.3.5] that there is a natural map of complexes of B-modules from thenormalized chain complexNLB/A associated to the simplicialB-module ΩP (B)/A⊗P (B)

B to what we called the truncated cotangent complex inducing an isomorphism onH0 and H1. In particular, the induced map on Ext1B(_ ,M) is an isomorphism forevery B-module M (Lemma 13.8.5), so Ext1B(LB/A,M) is the same whether LB/A

denotes the full" cotangent complex, or its truncation discussed in 6.

In particular, the statement of the Fundamental Theorem (Theorem 7.2.1) istrue regardless of whether LB/A means the full cotangent complex or the truncatedcotangent complex.


8.2. Basic properties. Here we summarize some of the basic properties of thecotangent complex. We begin with the analogs of Corollary 6.4.5, Lemma 6.5.1, andTheorem 7.1.1.

Theorem 8.2.1. Let f : X → Y be a map of schemes. Then the cohomology sheavesHn(LX/Y ) are quasi-coherent; they are coherent if f is a locally nite type map ofnoetherian schemes.

Proof. [Ill, II.2.3.7] Theorem 8.2.2. Given a pushout diagram of rings (or sheaves of rings)

A //

B

A′ // B′

where

TorA>0(A′, B) = 0

(this holds, for example, if at least one of A→ B, A→ A′ is at), the natural mapLB/A ⊗B B′ → LB′/A′ is a quasi-isomorphism.

Proof. [Ill, II.2.2] Theorem 8.2.3. If A → B → C are maps of sheaves of rings, there is a naturallong exact sequence of C-modules

· · · // H1(LB/A ⊗B C) // H1(LC/A) // H1(LC/B) //

// H0(LB/A)⊗B C) // H0(LC/A) // H0(LC/B) // 0

and for any C-module M there is a natural long exact sequence of C-modules

0 // Hom(LC/B,M) // Hom(LC/A,M) // Hom(LB/A ⊗B C,M) //

// Ext1(LC/B,M) // Ext1(LC/A,M) // Ext1(LB/A ⊗B C,M) // · · ·

Proof. See [Ill, II.2.1]. The proof is not particularly dierent from the proof ofTheorem 7.1.1 and is, in many ways, easier, since one needn't worry about the kindof boundary problems" caused by truncating which were the source of much of thesubtlety in the proof of Theorem 7.1.1. Theorem 8.2.4. Let f : X → Y be a morphism of schemes. Then the followingare equivalent:

(1) f is étale.(2) f is of locally nite presentation and LX/Y = 0.

Consider the following:

(1) f is smooth.


(2) The augmentation LX/Y → ΩX/Y is a quasi-isomorphism and ΩX/Y is locallyfree of locally nite rank.

(3) f is formally smooth.

Then (1) =⇒ (2) =⇒ (3) and these three conditions are equivalent if f is of locallynite presentation.

Proof. See [Ill, III.3.1]. For the rst statement, the non-obvious thing is to provethat f étale implies LX/Y = 0. Since an étale map is at, Theorem 8.2.2 saysthat π∗1LX/Y → Lπ2 is a quasi-isomorphism, where π1, π2 : X ×Y X ⇒ X are theprojections. The key point here is that the local nature of the cotangent complexcertainly ensures that Li = 0 when i : U → X is an open immersion. Since f isétale, the diagonal map i : X → X×Y X is an open immersion [EGA, IV.17.4.2] (inparticular it is at), so the transitivity triangle (Theorem 8.2.3) for

X → X ×Y X → X

(the maps are i and π2, so the composition X → X is the identity) implies that i∗Lπ2

is acyclic. But then applying i∗ to the rst-mentioned quasi-isomorphism shows thatLX/Y = i∗π∗1LX/Y is acyclic.

In the second map of the theorem, the non-obvious thing is to prove that whenf is smooth, H>0(LX/Y ) = 0. In light of the rst part of the theorem and thetransitivity triangle (Theorem 8.2.3), we know that the cotangent complex is étale-local in nature, so the desired result then follows from Lemma 8.1.1 and the factthat every smooth map of schemes is étale-locally (on domain and codomain) Specof a polynomial ring.

Remark 22. There are formally étale surjective ring homomorphisms A → B forwhich H2(LB/A) = 0.

8.3. Regular embeddings and l.c.i. morphisms. Here we recall some standardfacts about l.c.i. morphisms. For further reading, see [EGA, IV.16.9], Exposés VIIand VIII in [SGA6], Appendix B.7 in [Ful].

Denition 19. (c.f. [EGA, IV.16.9.2]) A closed embedding of schemes f : X → Yis called a regular embedding i the ideal of X in Y can be generated locally by aregular sequence (Denition 6). A morphism of schemes f : X → Y is called a localcomplete intersection (l.c.i.) morphism i f can be factored locally on X and Y asa regular embedding followed by a smooth morphism.

It is clear from the denition that a regular embedding is of locally nite pre-sentation. If I is the ideal of a regular embedding f : X → Y , then I/I2 is alocally free OX-module of locally nite rank (Lemma 4.2.1). An l.c.i. morphism is,in particular, of locally nite presentation.

It is fair to say that l.c.i. morphisms are a less well-behaved and less well-understood class of morphisms than, say, smooth morphisms. This despite the


fact that l.c.i. morphisms play a key role in many parts of algebraic geometry, par-ticularly in deformation theory and intersection theory, where they play a key rolein Grothendieck's Riemann Roch Theorem. For example, in [EGA] only about twopages are devoted to l.c.i. morphisms, while several hundred pages are devoted tosmooth morphisms. In [SGA6, VII] the Grothendieck school then introduces an al-ternative denition of an l.c.i. morphism which is in conict with the one in [EGA].However, in the world of nite type maps of noetherian schemes, the situation isbetterfor example the [EGA] and [SGA6] denitions agree.

One sense in which l.c.i. morphisms are not so well-behaved is that they arecertainly not closed under base change. For example, (Spec of) the map k[x] → kkilling x is a regular embedding (the paradigm example of a regular embedding isthe map A[x1, . . . , xn] → A killing the xifor any ring A), but if we pushout thismap along k[x]→ k[x]/x2, then we get the map k[x]/x2 → k killing x, which is not aregular embedding. However, l.c.i. maps and regular closed embeddings (dened asin [SGA6, VII]) are closed under at base change [SGA6, VII.1.5] and composition[SGA6, VII.1.7].

Theorem 8.3.1. If f : X → Y is an l.c.i. morphism of schemes, then LX/Y is ofperfect amplitude [−1, 0]i.e. is locally quasi-isomorphic to a complex L1 → L0 oflocally free OX-modules of locally nite rank (in particular Hn(LX/Y ) = 0 for n > 1,so LX/Y is quasi-isomorphic to its truncation). If f is a locally nite type morphismof locally noetherian schemes, the converse holds.

Proof. [Ill, III.3.2.6]

9. Deformation of schemes

Consider a commutative diagram of schemes

X

f

i // X ′

f ′

Y

j // Y ′

(9.0.1)

where i (resp. J) is a square-zero thickening with ideal I (resp. J). Then we havea natural map

f ∗J → I.(9.0.2)

This map is surjective i the diagram is cartesian.

9.1. Obstruction theory. Suppose now that we have a solid diagram of schemes

X

f

i // X ′

f ′

Y

j // Y ′

(9.1.1)

where j : Y → Y ′ is a square-zero thickening with ideal J .


Problem. Fix an OX-module I and a morphism of OX-modules u : f ∗J → I. Finda completion of (9.1.1) to a solid diagram of schemes (9.0.1) where i is a square zeroclosed embedding with ideal I such that the natural map (9.0.2) is given by the mapu. Such a completion will be called a solution to the problem.

By denition, the automorphism group of a solution is the group of automorphismsof the scheme X ′ which commute with i and f ′.

Theorem 9.1.1. There exists an element

ω ∈ Ext2X(LX/Y , I)

whose vanishing is necessary and sucient for the existence of a solution to the aboveproblem. If ω = 0, then solutions to the problem form a torsor under Ext1X(LX/Y , I).The automorphism group of any solution is canonically isomorphic to HomX(ΩX/Y , I).

Proof. The map i in (9.1.1) must be a homeomorphism on spaces (Lemma 2.1.1),so the problem is really just a problem about deforming maps of sheaves of ringson the space X. Set B := f−1OY , B′ := f−1OY ′ , J := f−1J (abuse of notation),C := OX , so f gives us a map B → C. Then LX/Y = LC/B. A solution to theproblem is the same thing (via the bijection taking C ′ to OX′) as a map of algebraextensions of the form

0 // I // C ′i // C // 0

0 // J

u

OO

// B′j //

f ′

OO

B

f

OO

// 0

(9.1.2)

The transitivity triangle (Theorem 7.1.1) for the maps of sheaves of rings

B′ → B → C

yields an exact sequence

0 // Ext1(LC/B, I) // Ext1(LC/B, I) // Ext1(LC/B′ , I) //

// Ext1(LB/B′ ⊗B C, I)δ // Ext2(LC/B, I)

(9.1.3)

of C-modules. The left map is injective because

Hom(LB/B′ ⊗B C, I) = Hom(ΩB/B′ , I)

= 0

since B′ → B is surjective. We have a commutative diagram

Ext1(LC/B′ , I) //

∼=

Ext1(LB/B′ ⊗B C, I)

∼=

ExalB′(C, I) // HomB(J, I)

(9.1.4)

where the right isomorphism is the sequence of natural isomorphisms

Ext1(LB/B′ ⊗B C, I) = HomC(J ⊗B C, I)

= HomB(J, I)


obtained from Lemma 13.8.3 (and the description of the cotangent complex LB/B′

of the square zero surjection B′ → B in Corollary 6.4.4) and the left isomorphism isthe Fundamental Theorem (Theorem 7.2.1). The bottom horizontal arrow in (9.1.4)is the composition of the natural map

ExalB′(C, I) → ExalB′(B, I)

and the isomorphism

ExalB′(B, I) = HomB(J, I)

of Theorem 3.3.1. Looking at how these maps are dened, we see that this map isjust the obvious map taking an algebra extension

0 // I // C ′i // C // 0

B′f ′

``BBBBBBBBfj

OO(9.1.5)

to the natural map f ′|J : J → I.

From this discussion, we see that a map of algebra extensions of the form (9.1.2)is the same thing as an algebra extension C ′ ∈ ExalB′(C, I) (as in (9.1.5)) whoseimage under the horizontal arrow in (9.1.4) is u ∈ HomB(J, I). By the exactness of(9.1.3) and commutativity of (9.1.4) the obstruction to nding such a C ′ is exactlyω := δ(u) (suppressing the right vertical isomorphism in (9.1.4)) and if ω = 0, theset of such C ′ is a torsor under Ext1(LC/B, I).

The statement about automorphisms of a given solution is elementary. First of allan automorphism of solutions is the same thing as a ring automorphism ϕ : C ′ → C ′

commuting with all the maps in (9.1.2). Given such a ϕ we obtain a B-linearderivation D : C → I by setting D(c) := ϕ(c′)− c′, where c′ ∈ C ′ lifts c (the choiceof lift is irrelevant, as one easily checks). Such a derivation is the same thing as anelement of HomC(ΩC/B, I). Conversely, given such a D, we obtain a ϕ by settingϕ(c′) := c′ +D(c) where c ∈ C is the image of c′ ∈ C ′ under C ′ → C. Remark 23. In the solid diagram (9.1.1), if f is at, then a completion f ′ is ati (9.0.2) is an isomorphism (Proposition 2.3.1). When f is at, the problem ofcompleting (9.1.1) as indicated with f ′ at reduces to the problem considered abovewhere u = Id : f ∗J → f ∗J . Given such a completion with f ′ at, pretty much anyproperty of the map f will be inherited by f ′ (Lemma 2.4.3). For example, anycondition on the geometric bers of f is inherited by f ′ because f and f ′ have thesame geometric bers since any map from a reduced scheme to Y ′ factors throughj : Y → Y ′.

9.2. Automorphisms of curves. Let k be an algebraically closed eld, C a smoothprojective curve of genus g ≥ 2. It is a standard exercise in Hartshorne to showthat the automotphism group of C (as a k-scheme) is nitelet us give a proofusing deformation theory and a little of the general theory of the Hilbert scheme.First of all, as part of the general quotient scheme machinery, Grothendieck showed


that the presheaf on Sch/k taking U to the group of automorphisms of U × Cover U is represented by a quasi-projective k-scheme Aut(C). Evidently then, theset of k-points Aut(C)(k) is the automorphism group of C as a k-schemesinceAut(C) is a nite-type group scheme over the algebraically closed eld k, we canshow that it has nitely many k-points by showing that the tangent space to Aut(C)at the identity (hence also at any other point) is trivial. To give an element of thistangent space is to give an element of Aut(C)(k[ϵ]/ϵ2) restricting to the identity onSpecC → SpecC[ϵ] (here C[ϵ] := C ×Spec k Spec k[ϵ]/ϵ

2). This in turn is the samething as an automorphism of the trivial solution

0→ ϵOC → OC [ϵ]→ OC → 0

of the lifting problem discussed in the previous section. By Theorem 9.1.1, the groupof such automorphisms is identied with

HomC(ΩC/k, ϵOC) = H0(C, TC)

where TC is the tangent sheaf of C over kthis group is trivial because g(C) ≥ 2,so the degree 2− 2g of TC is negative.

9.3. Moduli of nodal curves. Moduli spaces of nodal curves where introduced inthe famous paper [DM] of Deligne and Mumford. The moduli space of stable nodalcurves of a given genus can be used to compactify the moduli space Mg of smoothcurves of genus g ≥ 2. To illustrate some of the general deformation theoreticmachinery introduced in 9.1 we will treat the general problem of deformations ofnodal curves without regard to stability.

Denition 20. Let k be an algebraically closed eld. A k-scheme C is called a nodalcurve i C is proper over k, connected, purely one dimensional, and étale locallyisomorphic to a neighborhood of the origin in Spec k[x, y]/(xy) near each singularpoint of C. A map of schemes f : X → Y is called a (relative) nodal curve i itis at and proper and each of its geometric bers is a nodal curve in the previoussense.

For example, for any ring A, and any t ∈ A, Spec of the A-algebra map A →A[x, y]/(xy − t) is a nodal curve (except that it is nitely presented, though notproper). In fact, it can be shown that every nodal curve f : X → Y is étale locallyof this form in the sense that for any geometric point x of X, there is a commutativediagram

SpecA[x, y]/(xy − t) //

X

f

SpecA // Y

where the top horizontal arrow is an étale neighborhood of x and the bottom hori-zontal arrow is an étale neighborhood of f(x).

Let us compute the cotangent complex of A→ A[x, y]/(xy− t) =: B. First let usbegin by remarking thatB is freely generated (as anA-module) by 1, x, x2, . . . , y, y2, . . . ,


so in particular it is a at A-module. Any element b ∈ B can be uniquely writtenin the form

b = a+ a1x+ a2x2 + · · ·+ anx

n + b1y + · · ·+ bnyn(9.3.1)

for some a, a1, . . . , an, b1, . . . , bn ∈ A. Let us call this the standard form" of b.I claim xy − a is a regular element in A[x, y], so that (Spec of) A → B is l.c.i.(Denition 19). Indeed, suppose f(x, y) =

∑i,j ai,jx

iyj is a nonzero element ofA[x, y]. We have

(xy − t)f(x, y) =∑ij

(ai−1,j−1 − tai,j)xiyj

(adopting the convention that ai,j = 0 whenever one of i, j is negative). Chooseexponents m,n with m+n maximal with am,n = 0. Then the coecient of xm+1yn+1

in (xy − t)f(x, y) is am,n = 0, so (xy − t)f(x, y) = 0. The map A→ B comes witha presentation as a quotient of the free A-algebra A[x, y] by the ideal I = (xy − t),so its truncated cotangent complex (which is quasi-isomorphic to its full cotangentcomplex by Theorem 8.3.1 since A→ B is l.c.i.) is quasi-isomorphic to

[I/I2 → ΩA[x,y]/A ⊗A[x,y] B],(9.3.2)

where the boundary map is i 7→ di ⊗ 1 (Theorem 6.4.2). Since xy − t is a regularelement, I/I2 is the free B-module on the generator xy− t and ΩA[x,y]/A⊗A[x,y] B isthe free B-module on dx, dy. The boundary map in the complex sends xy − t to

d(xy − t) = ydx+ xdy

(note that dt = 0 since t ∈ A), so the complex (9.3.2) is isomorphic to the complexof free B-modules

[B → B⟨dx, dy⟩]where the boundary map takes b to ybdx + xbdy. I claim this map is injective.Indeed, if we write b in the standard form (9.3.1), then

ybdx+ xbdy =(a1t+ a2tx+ · · ·+ antx

n−1 + ay + b1y2 + · · ·+ bny

n+1)dx

+(b1t+ ax+ a1x

2 + · · ·+ anxn+1 + b2ty + · · ·+ bnty

n−1) dywhere we have written the coecients of dx, dy in the standard form; clearly thisvanishes only when b = 0. We have shown that H>0(LB/A) = 0, and that ΩB/A hasa free resolution of length one.

The above discussion shows, in particular, that every nodal curve over an alge-braically closed eld is l.c.i. (Denition 19).

Theorem 9.3.1. Given a solid commutative diagram of schemes (9.1.1) where f isa relative nodal curve and j : Y → Y ′ is a square-zero thickening of ane schemeswith ideal J , there exists a completion as indicated to a cartesian diagram where f ′

is a nodal curve.

Proof. By Theorem 9.1.1 and Remark 23 there is an obstruction

ω ∈ Ext2(LX/Y , f∗J)


whose vanishing is necessary and sucient for the existence of a completion asindictaed to a cartesian diagram where f ′ is atbut such a completion is auto-matically a nodal curve (Remark 23, Lemma 2.4.3), so it is enough to prove thatExt2(LX/Y , f

∗J) = 0. We do this in several steps:

Step 1: We claim H>0(LX/Y ) = 0, so that LX/Y → ΩX/Y is a quasi-isomorphism,and hence

Ext2(LX/Y , f∗J) = Ext2X(ΩX/Y , f

∗J).

We also claim that ΩX/Y locally admits a length one resolution by free OX-modulesof nite rank. For this, we appeal to the fact that a at map of locally nitepresentation with l.c.i. geometric bers is itself l.c.i.7 This means that, at leastlocally on X,Y , we can factor our nodal curve f as a regular embedding X → Mfollowed by a smoooth map M → Y . If I is the ideal of that regular embedding,then I/I2 is a locally-free OX-module of (nite rank) and ΩX/Y is the cokernel ofthe natural map

[I/I2 → ΩM/Y |X ],(9.3.3)

so we can obtain the desired resolution provided we prove this map is injective. Notethat this complex is also quasi-isomorphic to the truncated cotangent complex of f(Theorem 6.4.2), which is in turn quasi-isomorphic to the full cotangent complex off because f is l.c.i. (Theorem 8.3.1), so the injectivity statement we are about toprove will also establish the claim that H>0(LX/Y ) = 0.

To prove the injectivity, we use the fact that f is at, hence the map (9.3.3) is, inparticular, a map between quasi-coherent sheaves on X of locally nite presentationand at over Y , so by [EGA, IV.11.3.7], we can check injectivity on the bers (andhence on the geometric bers since eld extensions are faithfully at) of f . Butby at base change for the cotangent complex (Theorem 8.2.2), the restriction of(9.3.3) to a geometric ber Xy of f computes the cotangent complex of that ber,and we know this has no H1 by the discussion above.

Step 2: By the local-to-global spectral sequence for Ext, we reduce to proving thefollowing vanishings:

H2(X,H om(ΩX/Y , f∗J)) = 0(9.3.4)

H1(X,E xt1(ΩX/Y , f∗J)) = 0(9.3.5)

H0(X,E xt2(ΩX/Y , f∗J)) = 0(9.3.6)

The last vanishing (9.3.6) is immediate because E xt2(ΩX/Y , f∗J) = 0 in light of the

resolution property of ΩX/Y from Step 1.

Step 3: By the Leray spectral sequence for f and the fact that Y is ane, we have

Hn(X,F ) = Γ(Y,Rnf∗F )

7This is mentioned in 1 of [DM], but they give no proof or reference for this fact, nor could Ilocate one. If the schemes in question are locally noetherian, this is not so hard to prove, but I amunsure about the general case...


for every quasi-coherent sheaf F on X. The vanishing (9.3.4) is then clear fromthe fact that f is a proper map with one-dimensional bers, so R2f∗F vanishes forevery quasi-coherent sheaf F on X. The vanishing (9.3.5) holds for similar reasonsbecause f is smooth away from a closed subspace ofX of relative dimension zero overY , so ΩX/Y is locally free of nite rank away from a subspace of relative dimensionzero, so E xt1(ΩX/Y , f

∗J) is supported in relative dimension zero over Y , hence R1f∗of it vanishes.

10. Deformation of quotients

In this section we study the deformation theory of a surjection of quasi-coherentsheaves on a given scheme. We begin in 10.1 by studying a purely algebraic ab-straction of this problem, namely the deformation theory of an arbitrary surjectionof modules over an arbitrary sheaf of rings on an arbitrary space. We then introduceGrothendieck's Quot scheme in 10.2 and translate our abstract results into a defor-mation/obstruction theory for quotients in 10.3. As a rst application, we prove,using our deformation-theoretic results, that the Grassmannian is smooth (10.4).We also use our deformation theory of quotients to prove the smoothness of variousquotient schemes of vector bundles on curves (10.5).

10.1. Abstract deformation problem. Consider a square-zero surjection of rings(or sheaves of rings) R′ → R with kernel J . Fix an R′-module V ′ and let V :=V ′ ⊗R′ R = V ′/JV ′ ∈Mod(R). Assume that the natural sequence

V := (0→ J ⊗R V → V ′ → V → 0)(10.1.1)

is exactthe left map is j ⊗ v 7→ jv′ where v′ ∈ V ′ is any lift of v ∈ V (the choiceis irrelevant since J annihilates the kernel JV ′ of V ′ → V ).

Suppose f ′ : V ′ → F ′ is a map of R′-modules. Then, reducing mod J we obtaina map of R-modules f : V → F := F ′/JF ′ and a natural map

η : J ⊗R F → K(10.1.2)j ⊗ x 7→ jf ′(x′),

where we let K := JF ′ be the kernel of the natural surjection F ′ → F . We canpackage this as an exact diagram of R′-modules

0 // J ⊗R V //

η(J⊗f)

V ′

f ′

// V

f

// 0

0 // K // F ′ // F // 0

where the left and right vertical arrows are maps of R-modulesthroughout wethink of an R-module as an R′-module annihilated by J .


Problem: Fix a surjection f : V → F of R-modules and let N be its kernel so wehave an exact sequence

0→ N → V → F → 0.(10.1.3)

Fix an R-module K, and a map of R-modules u : J ⊗R F → K. The problem is tond a map f ′ : V ′ → F ′ of R′-modules reducing mod J to f such that the kernel ofF ′ → F is identied with K in such a way that the natural map (10.1.2) is givenby u. In other words, the problem is to complete the solid diagram of R′-modules

0 // J ⊗R V //

u(J⊗f)

V ′

f ′

// V

f

// 0

0 // K // F ′ // F // 0

(10.1.4)

as indicated to an exact diagram, for some F ′ ∈Mod(R′).

We will write f : V → F as shorthand for a solution (10.1.4).

Remark 24. As long as u is surjective, the f ′ in any solution will again be surjective(by the Snake Lemma and the surjectivity of u(J ⊗ f)). If F is at over R, then F ′

will be at over R′ i u is an isomorphism (Proposition 2.3.1).

Theorem 10.1.1. There is an obstruction ω ∈ Ext1R(N,K) whose vanishing isnecessary and sucient for the existence of a solution to the above problem. Ifω = 0, then the set of solutions to the above problem is a torsor under HomR(N,K).

Proof. Let L be the kernel of the composition V ′ → V → F . We have a commutativediagram

0

0

0

0 // J ⊗R V

i // L

p // N

// 0

0 // J ⊗R Vi //

V ′p //

V //

f

0

0 // F

F

// 0

0 0

(10.1.5)

with exact columns and rows. By pushing out the top row along u(J⊗f) we obtaina morphism of extensions

0 // J ⊗R V

u(J⊗f)

i // L //

N // 0

0 // K // M // N // 0

(10.1.6)


where M := K ⊕J⊗V L. Note that M is an R-module (i.e. JM = 0):

[jk, jl] = [0, jl] = [0, i(j ⊗ p(l))] = [u(j ⊗ f(p(l))), 0] = [u(0), 0] = 0,

so the bottom row denes a R-module extension. Let ω ∈ Ext1R(N,K) be theclass of this extension. Then a restatement of the theorem is that splittings of thebottom row in (10.1.6) (which are a pseudo-torsor under HomR(N,K)) correspondbijectively to solutions f : V → F to the problem.

Given a solution f : V → F , we obtain a commutative diagram

0 // J ⊗R V // N ′

// N //

0

0 // J ⊗R V //

u(J⊗f)

// V ′ //

f ′

V

f

// 0

0 // K // F ′ // F / / 0

(10.1.7)

with exact rows by pulling back the extension in the middle row along N → V . Notethat N ′ → V ′ is monic and N ′ ⊂ L ⊂ V ′, but N ′ need not be contained in the kernelof f ′. Given n ∈ N , choose a (local) lift n′ ∈ N ′. I claim that n 7→ [−f ′(n′), n′] ∈Mis well-dened (independent of the choice of lift n′ ∈ N ′), in which case it will clearlyprovide a splitting of the bottom row in (10.1.6). Indeed, if n′′ is another (local) lift,then n′′ = n′ + ϵ for some ϵ ∈ J ⊗R V and we have

[−f ′(n′ + ϵ), n′ + ϵ] = [−f ′(n′)− u(J ⊗ f)(ϵ), n′ + ϵ] = [−f ′(n′), n′] ∈M.

Conversely, notice that a splitting of the bottom row in (10.1.6) is the same thingas a map s : L→ K making the diagram

J ⊗R V

u(J⊗f)

i // L

swwwwwwwwww

K

(10.1.8)

commute. By pushing out the sequence dening L along s we obtain a commutativediagram

0 // J ⊗R V //

i

V ′ // V

f

// 0

0 // L

s

// V ′

// F // 0

0 // K // F ′ // F // 0

(10.1.9)

with exact rows, where we have set F ′ := K ⊕L V ′. The map from the top row tothe bottom row is evidently a solution to the problem since si = u(J ⊗ f).

We leave it to the reader to check that the two constructions above are inverse.


10.2. Quot schemes. Let f : X → Y be a map of schemes, E a quasi-coherentsheaf on X of locally nite presentation. The Quot functor is the functor

Q = Q(f, E) : (Sch/Y )op → Sets(10.2.1)

taking a Y -scheme U to the set of quotients π∗2E → F of π∗2E on U ×Y X such that:

(1) F is quasi-coherent and of locally nite presentation.(2) F has proper support over U via π1 : U ×Y X → U .(3) F is at over U via π1.

If f : X → Y is clear from context (e.g. if we are clearly working over some eld k),then we write Q(X,E) instead of Q(f, E).

Theorem 10.2.1. (Grothendieck) Let f : X → Y be a projective morphism ofnoetherian schemes, E a coherent sheaf on X. Then the Quot functor Q = Q(f, E)is is representable by a Y -scheme each of whose connected components is projectiveover Y . In particular, Q is of locally nite presentation over Y .

Proof. [Gro, 3.1]

More precisely: Fix an f -relatively ample invertible sheaf OX(1) on X. Thenfor a eld k, a k-point of Q consists of a point y : Spec k → Y and a quotient ofE|Xy → F of E|Xy on the projective k-scheme Xy = X ×Y Spec k. Then we canform the Hilbert polynomial

p(F ; t) :=∑n

χ(Xy, F (n))tn.

If we x a numerical polynomial p(t) ∈ Q[t], then the open-and-closed locus in Qconsisting of points of Q where the corresponding quotient has Hilbert polynomialp(t) is projective over Y .

Remark 25. It is clear from the denitions that the presheaf Q(f, E) commuteswith base change" in the following sense: If

X ′

f ′

// X

f

Y ′ // Y

is a cartesian diagram of schemes and we let E ′ denote the pullback of E to X ′ alongX ′ → X, then the diagram of presheaves

Q(f ′, E ′)

// Q(f, E)

Y ′ // Y

is also cartesian.


10.3. Obstruction theory. Now we rephrase the algebraic result of Theorem 10.1.1in geometric terms amenable to the theory of quotient schemes (10.2).

Consider a cartesian diagram of schemes

Xi //

f

X ′

f ′

Y

j // Y ′

(10.3.1)

where j is a square zero thickening with ideal J (hence i is also a square zerothickening with ideal I given by the inverse image ideal (f ′)−1J ⊆ OX′). As usual, weidentify the topological spaces ofX andX ′ (resp. Y and Y ′) via the homeomorphismsgiven by the horizontal arrows (Lemma 2.1.1) so that f and f ′ are the same mapon topological spaces. Let V ′ be an OX′-module (it does not matter in what followswhether V ′ is quasi-coherent, or of locally nite presentation, or anything like that).

We are going to make the following additional assumptions, which will be goodenough for our applications, though they can possibly be removed:

(1) V ′ is at over Y ′.(2) f ′ is at.

Set V := i∗V ′. Since V ′ is Y ′-at, the sequence

0→ f ∗J ⊗ V → V ′ → V → 0(10.3.2)

is exactit is obtained by tensoring the exact sequence

0→ f−1J → f−1OY ′ → f−1OY → 0

over f−1OY ′ with V ′. Since f ′ is at, the ideal I of i is equal to f ∗J , so

0→ f ∗J → OX′ → OX → 0(10.3.3)

is exact.

Fix a quotient q : V → F of OX-modules where F is at over Y .

Theorem 10.3.1. In the above situation, there is an element

ω ∈ Ext1X(N,F ⊗ f ∗J)

whose vanishing is necessary and sucient for the existence of a quotient q′ : V ′ →F ′ of OX′-modules with F ′ at over Y ′ and with i∗q′ = q. If ω = 0, then the set ofsuch quotients is a torsor under HomX(N,F ⊗ f ∗J).

Furthermore, if F is quasi-coherent (resp. of locally nite presentation), then forany such quotient q′ : V ′ → F ′, F ′ is quasi-coherent (resp. of locally nite presenta-tion).

Proof. We will apply Theorem 10.1.1 with R′ = OX′ , R = OX , J = f ∗J . (Note theexact sequence (10.3.3).) By Proposition 2.3.1, an OX′-module F ′ with i∗F ′ = Fwill be at over Y ′ i the natural sequence

0→ f ∗J ⊗ F → F ′ → F → 0


is exact. Hence a quotient q′ : V ′ → F ′ with i∗q′ = q and F ′ at is the same thingas a solution to the problem considered in Theorem 10.1.1 where the map u is theidentity map Id : f ∗J ⊗ F → f ∗J ⊗ F . This proves the rst part of the theorem.The second statement is Lemma 2.5.2. Corollary 10.3.2. Let f : X → Y be a at map of schemes, E a quasi-coherentsheaf on X at over Y . Let Q = Q(f, E) be the associated Quot functor. Given asquare-zero thickening of Y -schemes T → T ′ with ideal J and a map q : T → Qcorresponding to a quotient sequence

0→ N → π∗2E → F → 0

on T ×Y X, there is an obstruction ω ∈ Ext1T×Y X(N, π∗1J ⊗ F ) whose vanishing isnecessary and sucient for the existence of a lift q′ : T ′ → Q of q. If ω = 0, the setof liftings is a torsor under HomT×Y X(N, π∗1J ⊗ F ).

Proof. Apply the theorem with the diagram (10.3.1) given by

T ×Y X //

π1

T ′ ×Y X

π′1

T // T ′

and with V ′ = (π′2)∗E. Note that π′1 : T ′ ×Y X → T ′ is at since it is the base

change of f along T ′ → Y (V ′ is at over T ′ for similar reasons).

For the sake of concretenessand for the applications we have in mindlet usspecialize Corollary 10.3.2 to a concrete situation where we know Q is representableby a reasonable scheme.

Theorem 10.3.3. Let k be an algebraically closed eld, X a projective scheme overk, E a coherent sheaf on X. Let x be a k-point (i.e. a closed point) of the Quotscheme Q := Q(X,E) corresponding to an exact sequence

0→ N → E → F → 0(10.3.4)

of coherent sheaves on X. Then the tangent space to Q at x is given by

TxQ = HomX(N,F ).

Suppose, furthermore, that N is locally free and H1(X,N∨ ⊗ F ) = 0. Then there isa neighborhood of x in Q smooth over k.

Remark 26. The assumption that X is projective is only used to ensure thatthe Quot functor Q is represented by a locally nite type k-scheme; one couldalternatively just assume this representability and assume, say, that X is locallynite type over k.

Proof. The tangent space to Q at x is identied with the set of maps Spec k[ϵ]/ϵ2 →Q reducing to x mod ϵ. Such a map is the same thing as a lifting of (10.3.4) to aquotient sequence

0→ N ′ → E[ϵ]→ F ′ → 0(10.3.5)


on X[ϵ] := X ×Spec k Spec k[ϵ] with F ′ at over k[ϵ]/ϵ2. We have a canonical choiceof such a lifting given by

0→ N [ϵ]→ E[ϵ]→ F [ϵ]→ 0,

so Corollary 10.3.2 says that the set of such liftings is isomorphic as a k-vector spaceto HomX(N,F ⊗k ϵk) = HomX(N,F ).

For the furthermore," we will check the criterion in Corollary 5.9.3(5). We needto show that there is a lift in any diagram of k-schemes

SpecTq //

Q

SpecT ′q′

;;(10.3.6)

where T ′ → T is a small thickening of artinian k algebras with residue eld k and themap q takes the unique point of SpecT to x ∈ Q. Set XT := X ×Spec k SpecT andX ′T := X×Spec k SpecT

′ to ease notation. We have a cartesian diagram of k-schemes

X //

XT

// X ′T

// X

Spec k // SpecT // SpecT ′ // Spec k

(10.3.7)

where all but the right two horizontal arrows are closed embeddings. All of thevertical arrows are at and all the horizontal arrows induce isomorphisms on topo-logical spaces (Lemma 2.1.1). The ideal of the closed embedding XT → XT ′ is OX ,regarded as an OXT

-module via the closed embedding X → XT (because the idealof SpecT → SpecT ′ is k by denition of small thickening" and the vertical arrowsare at).

The map q corresponds to an exact sequence

0→ NT → ET → FT → 0(10.3.8)

of coherent sheaves on XT with FT at over T and reducing to (10.3.4) on the(unique) ber X ⊆ XT of XT → SpecT . Here ET denotes the pullback of E alongXT → X; it is at over T . I claim that NT is locally free (equivalently at). It is atover T since it is a kernel of the surjection ET → FT and ET and FT are at overT , so the claim follows from the berwise atness criterion (Lemma 4.5.6) becausewe assume N = NT ⊗OXT

OX is locally free.

A lifting q′ in (10.3.6) corresponds to a lifting of (10.3.8) to X ′T with F ′T atover T ′. By Corollary 10.3.2, such a lift will exist provided that some obstructionω ∈ Ext1XT

(NT , FT ⊗ OX) vanishes, so it is enough to show that the whole groupExt1XT

(NT , FT ⊗OX) vanishes. Since NT is locally free and NT and FT reduce to N


and F on the closed ber, this follows from the hypothesized H1-vanishing:

Ext1XT(NT , FT ⊗OX) = H1(XT , N

∨T ⊗ FT ⊗OX)

= H1(X,N∨ ⊗ F )

= 0.

10.4. Smoothness of the Grassmannian. The Grassmannian G = Gr(k, n) is anite type (even projective) scheme over SpecZ representing the presheaf taking ascheme X to the set of (isomorphism class of) exact sequences of OX-modules

0→ N → OnX → F → 0(10.4.1)

where N and F are locally free sheaves of rank k and n− k, respectively. Let us useTheorem 10.3.1 from the previous section to prove that G is smooth (over SpecZ).

Remark 27. Assuming only that the F in an exact sequence (10.4.1) is a at, locallynitely presented OX-module with n− k-dimensional bers (i.e. a locally free sheafof rank n− k by Lemma 4.4.3), the fact that N is locally free of rank k follows fromthe fact that a kernel of ats is at and a at sheaf of locally nite presentationis locally free (local nite presentation of N follows from Lemma 4.5.3). Note alsothat locally free of rank n − k" is the same as at, locally nitely presented, andbers have dimension n − k." This shows that the denition of the Grassmanniangiven above is the same as Quotk(SpecZ,Zn), as in 10.2. I preferred to do it likethis to avoid making use of Lemma 4.5.3. (A slight tradeo is that we have to worka little bit to prove that locally nite presentation of N passes to N ′ in a lifting of(10.4.1) over a thickening when F is lifted to a at F ′.

Since G is nite type, we can check formal smoothness. Consider a square zerothickening of ane schemes i : X → X ′ with ideal I and a map g : X → G. Weneed to show that g can be extended to g′ : X ′ → G. The map g corresponds to asequence (10.4.1) as above, and a lifting g′ corresponds to an exact sequence

0→ N ′ → OnX′ → F ′ → 0(10.4.2)

of OX′-modules reducing to (10.4.1) mod I with N ′ and F ′ locally free of rank k,n− k.

We apply Theorem 10.3.1 in the case where X = Y , X ′ = Y ′, and f , f ′ are theidentity maps (certainly then all of our additional assumptions" hold). Since N islocally free of nite rank and X is ane, we have

Ext1X(N,F ⊗ I) = H1(X,N∨ ⊗ F ⊗ I)

= 0,

so the theorem says we can nd an exact sequence (10.4.2) reducing mod I to(10.4.1) with F ′ a at OX′-module. Furthermore, the theorem says that this F ′ isof locally nite presentation, hence it is locally free of nite rank (Lemma 4.4.3)clearly it must then have rank n − k because the rank can be calculated on the


residue elds of points, and these are the same for both X and X ′. Since N ′ is thekernel of a map of at OX′-modules, it is also at, so to conclude that it is locallyfree (of rank k) we need only show that it is of locally nite presentation. Since itsreduction N = i∗N ′ = N ′/IN ′ is certainly of locally nite presentation, it sucesby Lemma 2.5.2 to prove that the natural sequence

0→ I ⊗N → N ′ → N → 0(10.4.3)

is exactthe only issue is the injectivity of the natural map η : I ⊗ N → N ′. Butη is part of a commutative diagram

0 // I ⊗N

η

// In //

I ⊗ F //

0

0 // N ′ // OnX′ // F ′ // 0

where the top row is exact since it is obtained by tensoring (10.4.1) over OX with Iand F is at over OX . Since In → On

X′ is certainly injective, η must be injective.

10.5. Smoothness of some Quot schemes. Theorem 10.3.3 can be used to provethe smoothness of various Quot schemes. This works especially well for locally freecoherent sheaves on curves. In this section we will work over a xed algebraicallyclosed eld k, though many of the results hold in greater generality.

Denition 21. By a curve, we mean a connected, smooth, projective k-scheme ofdimension 1.

Given a curve C and a coherent sheaf E on C (which will almost always belocally free...), we let Q(C,E) denote the corresponding quotient scheme (10.2).Each connected component of Q(C,E) is projective (Theorem 10.2.1) over k, soQ(C,E) is locally of nite type over k. In fact, if we let Qr,d(C,E) denote theopen-and-closed subscheme of Q(C,E) whose k-points are quotient sequences

0→ N → E → F → 0(10.5.1)

where N has rank r and degree −d (the sign here is convenient...), then Qr,d(C,E) isprojective. (The rank and the degree determine the Hilbert polynomial with respectto any ample line bundle on C.) We will refer to the integer

χ(X,N∨ ⊗ F ) := dimk H0(C,N∨ ⊗ F )− dimk H

1(C,N∨ ⊗ F )

as the expected dimension of Q(C,E) at the point (10.5.1). The expected dimensionis easily expressed in terms of r, d, and the rank and degree of E via the Riemann-Roch formula.

Lemma 10.5.1. Let C be a curve, E a locally free coherent sheaf on C,

0→ N → E → F → 0

an exact sequence of coherent sheaves on C. Then N is locally free and F is adirect sum of a locally free coherent sheaf and a coherent sheaf with zero-dimensionalsupport.


Proof. This follows immediately from the classication of nitely generated modulesover a PID because each stalk OC,x is a PID.

Theorem 10.5.2. Let C be a curve, E a locally free coherent sheaf on C, x ak-point (i.e. closed point) of Q(C,E) corresponding to an exact sequence (10.5.1)of coherent sheaves on C. Then H1(C,N∨ ⊗ F ) = 0 i Q(C,E) is smooth of theexpected dimension at x.

Proof. For =⇒ , apply Theorem 10.3.3, noting thatN is locally free by Lemma 10.5.1.For the converse, if Q = Q(C,E) is smooth of the expected dimension at x, then

dimk TxQ = χ(C,N∨ ⊗ F ).

But on the other hand, we have

dimk TxQ = dimk H0(C,N∨ ⊗ F )

by Theorem 10.3.3, so we must have H1(C,N∨ ⊗ F ) = 0.

We will see many examples where H1(C,N∨⊗F ) = 0, but yet Q(C,E) is smooth(though necessarily of unexpected" dimension).

Corollary 10.5.3. Fix any m,n ∈ N and let E denote the locally free coherentsheaf Om

P1 ⊕ OP1(1)n on P1 = P1k. Then for any exact sequence of coherent sheaves

(10.5.1) on P1 we have H1(P1, N∨ ⊗ F ) = 0. Consequently, Q(P1, E) is smooth ofthe expected dimension.

Proof. By Lemma 10.5.1 we can write F = G ⊕ T where T has zero-dimensionalsupport and G is locally free. By the classication of locally free coherent sheaveson P1, we can write G = ⊕iOP1(ai) for nitely many integers ai. We must haveai ≥ 0 for each i, because G is a quotient of E and E is generated by its globalsections, so G must be generated by its global sections. Similarly, we can writeN = ⊕jOP1(bj) for nitely many integers bj. Each bj must be at most 1 because Nis contained in E and there are no non-zero maps from OP1(b) to E if b > 1. SinceT has zero-dimensional support, so does N∨ ⊗ T , hence H1(C,N∨ ⊗ T ) = 0. Wend that

H1(P1, N∨ ⊗ F ) = H1(P1, N∨ ⊗G)

= H1(P1,⊕i,jOP1(ai − bj))

= 0

because H1(P1,OP1(c)) = 0 for c ≥ −1.

Remark 28. Tensoring quotients with an invertible sheaf L induces an isomorphismQ(X,E) ∼= Q(X,E ⊗ L) (for any X, E). A locally free coherent sheaf on P1 issometimes called balanced if it takes the form of the E in the above corollary aftertensoring with an invertible sheaf. The corollary shows that Q(P1, E) is smooth ofthe expected dimension for any balanced E on P1.


Remark 29. Corollary 10.5.3 holds over SpecZ, by essentially the same proof: Oneuses the smoothness criterion of Theorem 5.9.4, together with an argument muchlike the proof of Theorem 10.3.3, together with the fact that the H1-vanishing inCorollary 10.5.3 is valid over any eld.

Remark 30. The smooth projective schemes Q = Qr,d(P1,OnP1) were studied by

Stromme. Among other things, he showed that for each such Q, there is a GLn-torsor U over Q with U isomorphic to an open subspace of some ane space AN .He concludes that Q is rational and he computes its Chow ring. I am not awareof any signicant study of the smooth projective schemes Qr,d(P1, E) for arbitrarybalanced E.

Remark 31. For any locally free coherent sheaf E on P1 the quotient schemeQ1,d(P1, E) is smooth. The point is that there is only one rank 1 locally free sheafon P1 of degree −d and any nonzero map OP1(−d) → E is injective, so this Quotscheme is just a projective space:

Q1,d(P1, E) = PHomP1(OP1(−d), E).

However, this smooth Quot scheme will be of the expected dimension i E is bal-anced.

Corollary 10.5.4. Suppose the genus of the curve C is 1. Fix a positive integer n.For any short exact sequence

0→ N → OnC → F → 0

of coherent sheaves on C with N of rank one and of negative degree, we haveH1(C,N∨ ⊗ F ) = 0. Consequently, the Quot schemes Q1,d(C,On

C) are smooth ofthe expected dimension for all d > 0. If N has rank one and degree zero, thenN ∼= OC. We have Q1,0(C,On

C)∼= Pn−1, though the expected dimension of this Quot

scheme is zero. The Quot schemes Q1,d(C,OnC) are empty for d < 0.

Proof. We will leave the H1-vanishing as an exercise for the reader; it is an elab-oration on the analogous argument in the proof of Corollary 10.5.3. The secondstatement that N ∼= OC is obvious since the only degree zero intertible sheaf thatcan admit a non-zero map to On

C is OC . The rest is obvious.

Corollary 10.5.5. For any curve C and any locally free coherent sheaf E on C (ofrank r, say), the Quot scheme Qr,d(C,E) parameterizing coherent subsheaves of Eof full rank is smooth of the expected dimension.

Proof. In any exact sequence (10.5.1) where N and E have the same rank, F hasrank zero, hence has zero-dimensional support and the vanishing H1(C,N∨F ) = 0is trivial.

Corollary 10.5.6. For any curve C and any positive integer n, the Hilbert schemeHilbn C = Q1,d(C,OC) is smooth, projective, and of dimension n.


11. Hilbert schemes of points

Throughout this section we work over a xed algebraically closed eld k. Let Xbe a k-scheme. We will study the functor

X[n] := Hilbn X

from k-schemes to sets that takes a k-scheme U to the set of closed subschemesZ ⊆ U×X which are nite over U and have π1∗OZ a locally free OU -module of rankn. (For simplicity, let us in fact work implicitly with locally noetherian k-schemesso we can be a little less careful about the denition of the Hilbert functor. If wework with arbitrary k-schemes U we should add some more niteness hypotheses,like local nite presentation of Z.)

In particular, a k-point of X[n] is a closed subscheme Z ⊆ X nite over k withdimk H

0(Z,OZ) = n. Such a Z is set-theoretically supported at nitely many pointsx1, . . . , xm of X. The k vector space dimension of each of the local rings OZ,xi

is apositive integer ni and we have n = n1 + · · ·+ nm. Each of the local rings OZ,xi

liesbetween the local ring OX,xi

and its residue eld k and should thus be viewed as aninnitesimal thickening of xi in X. If each OX,xi

= k(xi) = k (i.e. Z is reduced),then we must have m = n. Evidently any choice of n distinct points of X yields ak-point of X[n] in this manner (we will make this rigorous in families" in 11.1).At the opposite extreme, when m = 1, we say that Z is punctual. Evidently anysuch Z is a nite disjoint union of punctual Z.

If X is quasi-projective over k, then Grothendieck's general theory of Hilbertschemes ensures that X[n] is representable by a quasi-projective scheme over k. Thisis one of the many variants of the Hilbert scheme Grothendieck discusses in [Gro, 4].Even if X is smooth and proper over k, the functor X[n] need not be representableby a k-scheme, though it is representable by an algebraic space in great generality[OS].8 Indeed, many results in this section hold in more generality, but in the interestof simplicity we will not work with algebraic spaces and we will pretty much stick toquasi-projective k-schemes. The assumption that k be algebraically closed is almostcertainly not necessary, but we keep it around to simplify the exposition.

For a quasi-projective k-scheme X, we will refer to the quasi-projective k-schemeX[n] as the Hilbert scheme of n points on X.

11.1. Conguration space. Recall the following general fact about the existenceof quotients (stated in much less than its full generality):

Theorem 11.1.1. (Gabriel) Let G be a nite, at group scheme over k acting ona k-scheme X. Assume that for all x ∈ X, the orbit Gx is contained in an aneopen subspace of X (this always holds if X is quasi-projective over k). Then there isa quotient q : X → Y in the category of k-schemes. More precisely, the action andprojection maps a, π : G × X ⇒ X have a coequalizer q : X → Y which coincides

8Hironaka's proper, non-projective smooth threefold X over C does not have X[2] representableby a scheme.


with the coequalizer taken in the category of ringed spaces and enjoys the followingproperties:

(1) The morphism q is integral.(2) If X is ane, then Y is ane.(3) The map (a, π) : G×X → X ×Y X is surjective.

If, furthermore, the action is free (i.e. (a, π) : G×X → X ×X is injective), then:

(4) The map q is nite and q∗OX is a locally free sheaf on Y of rank equal tothe rank of G.

(5) The map (a, π) : G×X → X ×Y X is an isomorphism.

Proof. See [SGA3, V.4.1]. The translation from group actions to groupoids is ex-plained in [SGA3, V.2(a)].

We often denote the Y in the above theorem by X/G. In particular, for a quasi-projective k-scheme X, we can form the quotient Xn := (Xn \ ∆)/Sn, where thesymmetric group Sn (viewed as a discrete group scheme, to be precise) acts bypermuting coordinates, and ∆ ⊆ Xn is the union of all the diagonals. The k-schemeXn is called the conguration space of n distinct, unordered points on X. Theobvious incidence locus Z ⊆ Xn×X determines an open embedding Xn → Hilbn X.If X is a variety (reduced and irreducible), then so is Xn, so the closure of Xn inHilbn X is an irreducible component called the distinguished component of Hilbn X.This distinguished component provides a compactication" of the congurationspace Xn.

If V ⊆ X is a locally closed embedding of quasi-projective k-schemes, we have amorphism

Hilbn V → Hilbn X(11.1.1)

representing the map of presheaves taking Z ⊆ U × V (a U -point of Hilbn V ) toZ ⊆ U ×X (one has to check that Z is still closed in the bigger space U ×X, whichfollows from the fact that Z is nite (hence proper) over U). If V ⊆ X is open, thisis an open embedding (if pressed to provide a careful proof of this fact, one provesthat this map is étale (by proving that it is formally étale) and radiciel, then quotes[EGA, IV.17.9.1]).

11.2. Fogarty's smoothness theorem. The goal of this section is to prove thatX[n] is smooth when X is a smooth surface (Theorem 11.2.4). This result wasoriginally proved by Fogarty [Fog] in the course of a study of the Hilbert scheme ofrelative dimension one closed subschemes for a map of schemes X → S.

[Discussion of dierent deformation-theoretic approaches to this....]

Lemma 11.2.1. Let A be a noetherian ring, x ∈ SpecA. If Ax is an integraldomain, then there is an f ∈ A \ x such that Af is an integral domain.


Proof. Let Pi be the set of minimal prime ideals of A. Then for any subset S ⊆ A,S−1Pi is the set of minimal prime ideals of the localization S−1A. Since Ax is adomain, 0 is its unique minimal prime ideal, so each (Pi)x is the zero ideal. Thismeans that for any Pi and any p ∈ Pi, there is some f = f(p) ∈ A \ x such thatfp = 0 in A. But A is noetherian so there are only nitely many Pi and each ofthem is nitely generated, so by multiplying together nitely many such f(p) wecan nd a single f ∈ A \ x such that fPi = 0 for all i. Then the set (Pi)f ofminimal prime ideals in Af consists soley of 0, so Af is an integral domain.

Lemma 11.2.2. Let A be a regular local k-algebra of dimension 2 with residue eldk. Let I ⊆ A be an ideal of A such that B := A/I has nite dimension n as a kvector space. Then dimk HomA(I, B) = 2n.

Proof. Let m be the maximal ideal of A, K the fraction eld of A. Since B hasnite dimension as a k vector space, we must have mN ⊆ I for large enough N . Inparticular, mNB = 0, so B ⊗A K = 0. Since the 2-dimensional regular local ring Ahas homological dimension 2 (Theorem 4.3.1), we can extend the natural surjectionA→ B to a length 2 free resolution

0→ Ar → Am → A→ B → 0(11.2.1)

of B. Since K is at over A, this becomes an exact sequence of K-vector spacesafter applying _ ⊗A K, so since B ⊗A K = 0, we must have m = r + 1. ApplyingHomA(_ , B) to the free resolution of B in (11.2.1), we get a complex

B → Br+1 → Br(11.2.2)

whose cohomology groups compute ExtiA(B,B) for i = 0, 1, 2. The complex (11.2.2),viewed as a complex of (nite dimensional!) k vector spaces, has Euler characteristic

dimk B − dimk Br+1 + dimk B

r = 0.

Since this Euler characteristic is the same as the alternating sum of the dimensionsof its cohomology, we nd

2∑i=0

(−1)i dimk ExtiA(B,B) = 0.

On the other hand, HomA(B,B) = B has dimension n as a k vector space and wehave an isomorphism of k-vector spaces

HomA(B,B) = Ext2A(B,∧2ΩA/k ⊗B)∨(11.2.3)


(where the superscript ∨ is the k-linear dual) by Serre Duality.9 Of course ΩA/k∼= A2

(non-canonically), so (11.2.3) implies that dimk Ext2A(B,B) = n, so we conclude that

dimk Ext1A(B,B) = 2n.

It now suces to show that

Ext1A(B,B) = HomA(I, B).

For this we need only apply HomA(_ , B) (and its derived variants) to

0→ I → A→ B → 0

and note that the natural map HomA(B,B) → HomA(A,B) is an isomorphism,hence the connecting map

HomA(I, B)→ Ext1A(B,B)

is an isomorphism because Ext1A(A,B) = 0.

Lemma 11.2.3. Let X be a connected scheme of nite type over k. Assume thatthere is a closed point x0 of X such that X is smooth at x0. Set n := dimk Tx0X.Assume furthermore that dimk TxX = n for every closed point x of X. Then X issmooth over k.

Proof. Since X0 is smooth at x0, OX,x0 is regular (Corollary 5.8.2), hence a domain(Theorem 4.3.4), so there is some neighborhood U of x0 in X which is a variety(Lemma 11.2.1). Since OX,x0 is regular and

n = dimk Tx0X = dimk mx0/m2x0,

OX,x0 has Krull dimension n, so the closure X0 of U in X is a subvariety of Xof dimension n. Then the Krull dimension of OX,x at each closed point x ∈ X0

is n and hence the hypothesis dimk TxX = n ensures that OX,x is regular, so X0

is smooth (Corollary 5.8.2 again). But we have to have X0 = X, otherwise, byconnectedness, there would be some irreducible component Y of X0 distinct fromX0 and intersecting X0, but then the local ring at a closed point of Y ∩X0 couldn'tbe a domain (Lemma 11.2.1 again), so in particular it couldn't be regular.

Theorem 11.2.4. (Fogarty) Let X be a smooth, quasi-projective surface over k.Then X[n] is a smooth k-scheme of dimension 2n, connected when X is connected,projective when X is projective.

9I have in mind some appropriate form of Serre Duality for modules supported at the closed pointof a local ring, but one could reduce this to good-old Serre duality for, say P2

k, as follows: Firstnotice that the statement we are trying to prove doesn't really depend on A: We could replaceA by A/mN for some very large N . The ring A/mN is independent of the particular A we workwith because all these A have the same completion, namely k[[x, y]]. In particular, we could takeA to be the local ring of a closed point x ∈ P2

k, then view B as a coherent sheaf on P2k supported

at x and apply Serre duality to P2k. All the Ext's computed on P2 reduce immediately by the

local-to-global spectral sequence to the Ext's we want.


Proof. The projectivity statement is part of Grothendieck's Theorem 10.2.1. Theconnectedness statement will be proved in the next section (Lemma 11.3.1 plusLemma 11.3.4), so we can use Lemma 11.2.3 (applied with x0 a point of the smoothopen subspace given by conguration space Xn ⊆ X[n]) to reduce to showing thatTyX[n] has dimension 2n for every closed point y of X[n]. Such a closed point ycorresponds to a closed subscheme Z ⊆ X with dimkOZ = n. By Theorem 10.3.3,we have

TyX[n] = HomX(I,OZ),

where I ⊆ OX is the ideal sheaf of Z in X, so it suces to show that HomX(I, Z)has dimension 2n. Let x1, . . . , xm ∈ X be the (necessarily closed) points of X whereZ is supported and let ni := dimkOZ,xi

, so we have n = n1+ · · ·+nm. We concludeby noting that

HomX(I,OZ) = ⊕mi=1HomOX,xi

(Ixi,OZ,xi

)

has dimension 2n1 + · · · + 2nm = 2n by Lemma 11.2.2. (Note that each OX,x is aregular local ring of dimension 2 by Corollary 5.8.2).

11.3. The punctual Hilbert scheme. Let X be a quasi-projective k-scheme,x ∈ X a closed point. Fix N ∈ N and let ηN(x) := SpecOX,x/m

N+1x be the N th

innitesimal neighborhood of x in X. The closed embeddings

· · · → ηN(x) → ηN+1(x) → · · · → X

induce closed embeddings

· · · → ηN(x)[n] → ηN+1(x)[n] → · · · → X[n].

For any xed n, we can nd an N so that every ideal I ⊆ OX,x with dimkOX,x/I = ncontains mN

x . In other words, we can nd a single N so that every closed subschemeZ ⊆ X supported set theoretically at x and with dimkOZ = n is actually containedin ηN(x). The sequence of closed embeddings above therefore eventually stabilizersand it makes sense to set

X[n]x := lim−→ηN(x)[n] : N ∈ N.

Since ηN(x) is certainly projective over k (it is even nite), the general theory of theHilbert scheme (Theorem 10.2.1) ensures that X[n]x is projective over k.

Since X[n]x manifestly depends only on the k-algebra OX,x/mNx for suciently

large N , it is clear that all the X[n]x depend only on the complete local ring OX,x.We are thus justied in writing Hilbn OX,x for X[n]x.

In particular, for any smooth closed point x ∈ X, X[n]x = Hilbn k[[x1, . . . , xd]],where d is the dimension of the regular local ring OX,x.

Lemma 11.3.1. Let X be a smooth, connected, quasi-projective k-scheme of dimen-sion d. If the punctual Hilbert scheme Hilbn k[[x1, . . . , xd]] is connected, then X[n]is connected.


Proof. Since X is smooth, connected, and quasi-projective it must be a variety. Fixa closed point p ∈ X. It suces to show that every closed subscheme Z ⊆ X withdimkOZ = n can be connected to" (i.e. lies in a at family over a connected basewith) a closed subscheme set-theoretically supported at p. Let y1, . . . , ym be thepoints in the support of Z which are not equal to p. We proceed by induction onm. If m = 0, we're done, so assume m > 0. Set y := y1 and write Zy ⊆ X for thecomponent of Z supported at y. Since X is smooth, we can nd an open (necessarilyconnected) neighborhood U of y in X and an étale map f : U → Ad

k. We can alsond an open neighborhood V of p in X and an étale map g : V → Ad

k. By removingy2, . . . , ym from U and V if necessary, we can assume that y is the only point ofSuppZ contained in U and that SuppZ is disjoint from V , except for the possibilityof Z being supported at p, which is okay with us. Since X is irreducible, we cannd a closed point z ∈ U ∩ V not equal to x.

Our chosen étale map f provides a canonical identication

OU,x = OAdk,f(x)

= k[[x1, . . . , xd]]

for every closed point x ∈ U . Using this canonical identication, we can translate Zy

from y to any other point x ∈ U to obtain a new closed subscheme Zx supported atx (and isomorphic to the original Zy as an abstract k-scheme). This gives us a mapfrom the connected open subset U to X[n] taking x ∈ U to the closed subschemeZ(x) ⊆ X obtained from Z by replacing Zy with Zx. (Since y is the only point of Uin the support of the original Z, there is no possibility of trying to pile Zy on top ofanything...) We can thus connect Z to some new closed subscheme Z ′ ∈ X[n] withthe same m as the m for Z, but now supported at z ∈ V .

Now we just need to connect this Z ′ to a point with smaller m. We can do thesame thing as before using the étale map g to get a map V \ p → X[n] takingx to the closed subscheme Z(x) ⊆ X obtained by replacing Z ′z with Z ′x. It is notso clear that we can extend this map to p because it is not so clear how to pile Z ′zon top of the component Zx of Z at x (when this is non-empty). But we can justappeal to properness of the Hilbert scheme as follows: Pick a smooth connectedcurve C and a non-constant map h : C → V taking some point 0 ∈ C to p ∈ X.Passing to an open subset of C if necessary, we can assume 0 = h−1(p), so we havea map h : C \ 0 → V \ p. Composing with our previously constructed mapV \p → X[n] yields a map C \0 → X[n]. Now, X may not be projective, but itis quasi-projective, so it is an open subspace of a projective k-scheme X, and X[n]is projective, so we can complete our map to C → X[n]. For c ∈ C not equal tozero, this map takes c to a closed subscheme supported at c, y2, . . . , ym, so it is clearjust on topological grounds that this map has to take 0 to some closed subschemeZ(0) ⊆ X supported at 0, y2, . . . , ym. In particular we must have Z(0) ⊆ X and thisshows that we can connect Z to a new closed subscheme Z(0) supported at onlym− 1 points other than p, so we're done by induction.


Lemma 11.3.2. Let X be a proper scheme over k equipped with an action of thetorus T = (Gm)

nk . Let X

T be the xed locus. Then for every point x ∈ X,

Tx ∩XT = ∅.Here Tx ⊆ X is the T -orbit of x and Tx is its closure in X (sometimes regarded asa k-variety by giving it the reduced induced closed subscheme structure).

Proof. For x ∈ X, let Sx ⊆ T be the stabilizer of x, so that Tx ∼= T/Sx. A quotientof a torus is again a torus, to T/Sx is again a torus. We proceed by induction onthe dimension of this torus. If it is zero-dimensional, then it is the trivial group,so we must have Sx = T , hence x ∈ Tx ∩ XT and we are done. If d > 0, thenthe torus Tx ∼= T/Sx is not proper, so it cannot be closed in the proper scheme X,hence we can nd y ∈ Tx \ Tx. This implies that Ty ⊆ Tx \ Tx and hence alsothat Ty ⊆ Tx \ Tx. Then Ty is contained in a proper closed subset of the varietyTx, hence dimTy = dimTy is smaller than d, so Ty ∩ XT = ∅ by induction, butTy ∩XT ⊆ Tx ∩XT , so we're done.

We won't actually make direct use of the next lemma, but it seems to be worthmentioning at this point:

Lemma 11.3.3. Let X be a proper scheme over k equipped with an action of thetorus T = (Gm)

nk . Let X

T be the xed locus. Then X is smooth i OX,y is a regularlocal ring for each closed point y ∈ XT .

Proof. The condition is clearly necessary for smoothness; to show that it is sucient,it suces to show that the local ring OX,x at an arbitrary closed point x ∈ X isregular (Corollary 5.8.2). By Lemma 11.3.2, there is a closed point y ∈ Tx ∩ XT .By openness of the smooth locus (Corollary 5.6.3), there is an open neighborhoodU of XT in X which is smooth (over k, so the local rings at points of U are regular).Since y ∈ Tx, the open neighborhood U of y in X must contain some point x′ ∈ Tx.But x is taken to x′ by some automorphism of X, so OX,x

∼= OX,x′ and OX,x′ isregular since x′ ∈ U . Lemma 11.3.4. The punctual Hilbert schemes Hilbn k[[x, y]] are connected for alln.

Proof. Set H := Hilbn k[[x, y]]. The torus T = G2m acts on A2 by rescaling the

coordinates x, y. This induces a T -action on A2[n]. Since the T -action xes theorigin, the punctual Hilbert scheme H = A2[n]0 ⊆ A2[n] is invariant under thisaction, so we have an induced action of T on H. The xed points of this actioncorrespond to monomial ideals I ⊆ k[[x, y]] =: A with dimk A/I = n. (Note that allthe T xed points of A2[n] happen to lie in H, but the T action on A2[n] is not ourpresent concern.) A typical such monomial ideal I correponds to a partition λ of n,as follows: Write λ = (λ0, . . . , λl) with λ0+ · · ·+λl = n and λ0 ≥ λ1 ≥ · · · ≥ λl > 0.The monomial ideal Iλ associated to λ is generated by

xλ0+1, xλ1+1y, . . . , xλl+1yl, yl+1.


It is helpful to picture such a partition λ and the corresponding ideal Iλ by drawinga small box around (m,n) ∈ R2 for each (m,n) ∈ N2 for which xmyn is not in Iλ.These boxed monomials form a basis for the quotient Bλ := A/Iλ. The monomialsin question are

yl, xyl, x2yl, . . . , xλlyl

yl−1, xyl−1, x2yl−1, . . . , xλl−1yl−1

...y, xy, x2y, . . . , xλ1y

1, x, x2, . . . , xλ0 .

SinceH is projective, Lemma 11.3.2 ensures that each point ofH can be connectedto a torus xed point, so the issue is to connect two monomial ideals. Let us showthat we can connect a given Iλ to the monomial ideal J := (xn+1) correspondingto the partition λ0 = n of length one. It suces to show that we can move therightmost box on the top row (xλlyl) to the right of the bottom row (xλ0+1)". Thatis, we can connect Iλ for any partition λ = (λ0, . . . , λl) with l > 0 to Iµ, whereµ = (λ0 + 1, . . . , λl − 1). Repeatedly moving boxes in this manner, we'll eventuallyget them all lined up on the bottom row. We let I := Iλ ∩ Iµ. Note that A/I hasdimension n + 1: a basis for this quotient is given by the pile of boxes where weleave the right box on the top of λ where it is, and we stick another box on thebottom row of λ. Now we can dene a map P1 → H by taking [s : t] to the idealI[s : t] generated by I and sxλlyl + txλ0+1. It is easy to see that each quotientA/I[s : t] has dimension n and that I[1 : 0] = Iλ, I[0 : 1] = Iµ, so we have a mapP1 → Hilbn k[[x, y]] connecting Iλ and Iµ as desired.

12. Pathologies

In this section we mention some standard examples of various pathological" be-havior in deformation theory. I will often give the examples without going intodetails of the proofs, since the purpose of these examples is mainly to illustrate thatnot everything one hopes to be true is actually true.

12.1. Rigid schemes. This section and the next are based very closely on thecorresponding sections in [H2].

Proposition 12.1.1. Let X be a nite type scheme over a eld k. The followingare equivalent:

(1) Ext1(LX/k,OX) = 0.(2) Every at family of schemes over k[t]/t2 reducing to X mod t is isomorphic

to the trivial family X ×k k[t]/t2.

(3) For every Artinial local k-algebra (A,m) with residue eld k, every at familyof schemes over A reducing mod m to X is isomorphic to the trivial familyX ×k A.


Denition 22. A scheme X of nite type over a eld k satisfying the equivalentconditions of Proposition 12.1.1 is called rigid.

For example, P1 := P1k is rigid since

Ext1(LP1/k,OP1) = Ext1(ΩP1/k,OP1)

= H1(P1, TP1)

= 0.

The rst innitesimal neighborhood of the zero section in the total space of OP1(−1)is an example of a (non-reduced, hence singular) rigid scheme (Exercise 18). Wewill see later (Proposition 12.2.3) that a singular rigid scheme is not smoothable.

12.2. Formally smoothable schemes. Here we give a summary of Hartshorne'stheory of formal smoothability and explain how it can be used to prove that variousschemes are not smoothable.

Throughout this section we work over a xed eld k. It is not particularly im-portant that k be algebraically closed or of characteristic zero, but the reader willlose nothing by assuming k = C. Throughout, X, Y, . . . will denote nite typek-schemes.

Denition 23. We say that X and Y are deformation equivalent i there are: asmooth connected curve C over k, a nite type k-scheme Z, a at map of k-schemesf : Z → C, and k-points x, y of C such that the ber of f over x (resp. y) isisomorphic to X (resp. Y ). We say that X is smoothable i X is deformationequivalent to Y for some smooth Y .

In the above denition, one can replace smooth connected curve" with reasonableconnected k-scheme" because any two k-points of any reasonable connected k-schemeB are the images of two k-points of a smooth connected curve C under a mapC → B. There are other possible variants of the denition of smoothable": onecould instead say that X is smoothable i X is the closed ber of a at family overa DVR with residue eld k whose generic ber is smooth (or to be concrete onecould take the DVR to be k[[t]]). The subtle dierences between these denitionswill not be particularly important in what follows: our intention is only to givesome examples of schemes which are not smoothable (in any of the aforementionedsenses).10

Denition 24. Let An := k[t]/tn+1. A compatible family is a sequence of at, nitetype maps X• = Xn → SpecAn : n ∈ N such that Xn+1 reduced mod tn+1 to Xn

10The dierence between a at family over k[[t]] and one over a smooth curve over k is a subtleissue of algebraizability that we needn't go into at this point.


for each n ∈ N. That is, there are cartesian diagrams

Xn

// Xn+1

SpecAn

// SpecAn+1

where the bottom arrow is Spec of the obvious quotient map An+1 → An. We referto the nite type k-scheme X0 as the ber of X•.

A compatible family" is a basically a poor man's way of saying a at formalscheme over Spf k[[t]]." Note that the reduction maps Xn → Xn+1 are squarezero thickenings, so they are isomorphisms on topological spaces; one gets a for-mal scheme at over Spf k[[t]] (the one point space with sheaf" of rings k[[t]]) byendowing the common topological space of the Xn with the sheaf of k-algebraslim←−OXn (this is a sheaf of k[[t]]-algebras).

Denition 25. A nite type k-scheme X0 is called formally smoothable i there isa compatible family X• with the following property: There is an N ∈ N such thattN E xt1(LXn/An ,G ) = 0 for every n ∈ N and every coherent sheaf G on Xn.

The whole point of the above denition is the next theorem, plus the fact that itis much easier in practice to prove that a scheme is not formally smoothable thanit is to prove that the scheme is not smoothable.

Theorem 12.2.1. (Hartshorne) Let X be a nite type scheme over a eld k. IfX is smoothable then X is formally smoothable.

Denition 26. (Hartshorne) Let A be a noetherian ring. A map of abeliancategories F : Mod(A) → Mod(A) is called coherent i it is isomorphic to thecokernel of functors corepresented by nitely generated A-modules. In other words,F is coherent i there is a map f : Q → P of nitely generated A-modules, and anatural isomorphism of A-modules

F (M) = Cok(f ∗ : HomA(P,M)→ HomA(Q,M))

for each A-module M .

The notion of coherent functor is interesting precisely because the Yoneda em-bedding

Mod(A)op → End(Mod(A))

P 7→ HomA(P, _ )

(f : Q→ P ) 7→ (f ∗ : HomA(P, _ )→ HomA(Q, _ ))

from Mod(A) to the category of abelian endofunctors of Mod(A) (like all othersorts of Yoneda embedding) preserves inverse limits but not direct limits.

Lemma 12.2.2. Let A be a noetherian ring, t ∈ A. For a coherent functor F :Mod(A)→Mod(A), the following are equivalent:


(1) There is an n ∈ N such that tnF (M) = 0 for every A-module M .(2) For every nitely generated A-module M , there is an n ∈ N such that

tnF (M) = 0.

Proposition 12.2.3. A rigid nite type k-scheme is not formally smoothable, andis hence not smoothable by Theorem 12.2.1.

12.3. Nonsmoothable schemes. Mumford's example of a nonsmoothable curvesingularity, Pinkham's examples of nonsmoothable normal surface singularities, zero-dimensional subschemes of A3 that aren't smoothable

12.4. Iarrobino's Example: Hilb96 A3. Iarrobino observed that Hilb96 A3 is notirreducible, for rather trivial dimension-counting reasons, as follows: First of all, wehave an open embedding

((A3)96 \∆)/S96 → Hilb96 A3

from the conguration space ((A3)96 \ ∆)/S96 parameterizing unordered 96-tuplesof distinct points on A3 into the Hilbert scheme Hilb96 A3 of length 96 closed sub-schemes of A3. The closure in Hilb96 A3 of this open subspace is, by denition, thedistinguished component. It is an irreducible component of Hilb96 A3 of dimension3 · 96 = 288.

Let A := k[x, y, z] be the coordinate ring of A3 and let m = (x, y, z) be the idealdening the origin. For any 24-dimensional k-linear subspace V ⊆ m7/m8, considerthe ideal IV := I + m8 of A, and the corresponding quotient A → BV := A/IV .The quotient BZ is supported at the origin of A3 (it is annihilated by m8). Notethat mn has the monomials in x, y, z of degree ≥ n as a k-basis, so m7/m8 hasdimension equal to the number of degree 7 monomials in x, y, zi.e. 36. Similarly,the dimension of BV as a k vector space is the number of monomials in x, y, z ofdegree at most 7 (i.e. the dimension of A/m8) minus 24, which is 96.

The choice of V is the choice of a 24-dimensional subspace in a 36-dimensionalvector spacei.e. a point in the Grassmannian G = Gr(24,m7/m8) = Gr(24, 36),which has dimension 24 ·12 = 288. Since V can be recovered from IV by the formulaV = IV /m

8, V 7→ IV denes a monomorphism Gr(24, 36) → Hilb96 A96. But wealso have the freedom to translate our construction along any vector in A3 (movinga subscheme supported at the origin in A3 to a subscheme supported at some otherpoint (a, b, c) by translation), so actually we have an inclusion A3 × Gr(24, 36) →Hilb96 A3. We have thus produced some subspace of Hilb96 A3 of dimension 288+3 =291; since this subspace cannot be contained in the distinguished component ongrounds of dimension, Hilb96 A3 cannot be irreducible.

13. Homological Algebra

This appendix is mostly concerned with various Ext groupsparticularly sheafExt and hyper Ext. The results are generally elementary and fairly self-contained.The reader is assumed to be familiar with homological algebra at the level of any


standard textbook on the subject, Grothendieck's Tôhoku article [T], or at least thebeginning of Chapter III in [H].

13.1. Yoneda Ext. At various points in the text we will make use of Yoneda'sdescription of Ext groups in an abelian category A. The results of this section areadapted from [Y]. Proofs omitted below can be found therein.

For objects A and B of A and a positive integer n, we let YExtnA(A,B) (or justYExtn(A,B) if A is clear from context) denote the set11 of exact sequences

0 // B // En// · · · // E1

// A // 0(13.1.1)

in A. We often denote such a sequence E. A morphism from such a sequence E toanother such sequence E ′ is a commuative diagram

0 // B // En

// · · · // E1

// A // 0

0 // B // E ′n // · · · // E ′1 // A // 0

(13.1.2)

in A. We let ExtnA(A,B) (or just Extn(A,B)) denote the quotient of YExtn(A,B)by the smallest equivalence relation containing the pairs (E,E ′) for which there isa morphism E → E ′. Explicitly, E,E ′ ∈ YExtn(A,B) have the same image inExtn(A,B) i there is a sequence

E = E0, . . . , Em = E ′

of elements of YExtn(A,B) such that, for i = 1, . . . ,m, there is a morphism Ei−1 →Ei or a morphism Ei → Ei−1.

Note that when n = 1, the map E1 → E ′1 in any diagram (13.1.2) is automaticallyan isomorphism by the Five Lemma, so the relation given by the pairs (E,E ′) ofelements in YExt1(A,B) for which there is a morphism E → E ′ is already anequivalence relation. Hence Ext1(A,B) is the group of short exact sequences

0→ B → E → A→ 0

up to the evident notion of isomorphism.

The set Extn(A,B) has the structure of an abelian group under Baer sum. Theadditive identity 0 ∈ Extn(A,B) is the equivalence class of

0→ B → B → 0→ · · · → 0→ A→ A→ 0.(13.1.3)

(When n = 1, the additive identity is the equivalence class of the split short exactsequence.) An A-morphism f : B → B′ induces a group homomorphism

Extn(A,B) → Extn(A,B′)(13.1.4)

by taking (the equivalence class of) an exact sequence (13.1.1) to the exact sequence

0 // B′ // B′ ⊕B En// En−1 // · · · // E1

// A // 0.

11There is an obvious set-theoretic issue here which we will ignore.


In fact, Extn(A, _ ) denes a map of abelian categories A→ Ab. There is a dual"contravariant functoriality in A.

Given an exact sequence

0→ B′ → B′′ → B → 0(13.1.5)

in A and a positive integer n, there is a natural map

Extn(A,B) → Extn+1(A,B′)(13.1.6)

taking (13.1.1) to the exact sequence

0 // B′ // B′′ // En// · · · // E1

// A // 0.

The maps (13.1.6) furnish the higher" connecting maps in a long exact sequence

0 // Hom(A,B′) // Hom(A,B′′) // Hom(A,B)δ //

// Ext1(A,B′) // Ext1(A,B′′) // Ext1(A,B) // · · ·(13.1.7)

naturally associated to the short exact sequence (13.1.5). The rst connecting map δin the sequence (13.1.7) has a slightly dierent construction: it takes f ∈ Hom(A,B)to the short exact sequence in Ext1(A,B′) obtained by pulling back (13.1.5) alongf . This endows Ext•(A, _ ) with the structure of a δ-functor

Ext•(A, _ ) : A→ Ab.(13.1.8)

We will prove in 13.3 that the δ-functor (13.1.8) is universal [H, 3.1] (withoutany assumptions on the abelian category A), hence the funtors Extn(A, _ ) deserveto be called the right derived functors of Hom(A, _ ). Before doing this, we rstnote that if B is an injective object of A, then Extn(A,B) = 0 for all n > 0. Indeed,if B is injective, any E ∈ YExtn(A,B) is isomorphic to an element of the form

0→ B → B ⊕ C → En−1 → · · · → E1 → A→ 0(13.1.9)

and there is an evident morphism from (13.1.9) to (13.1.3). Consequently, when Ahas enough injectives, the δ-functor (13.1.8) is eaceable, hence is universal by abasic result of Grothendieck [T, II.2.2.1]. In this situation, then, (13.1.8) coincideswith the universal δ-functor dened by

B 7→ Hi(Hom(A, I•)),(13.1.10)

where I• is any injective resolution of B in A (the right hand side of (13.1.10) doesnot depend on the choice of such a resolution). We will prove universality of (13.1.8)in 13.3 without the assumption of enough injectives in A.

The results of this section have formal duals" obtained by looking at the oppositeabelian category Aop. For example, the exact sequence (13.1.5) also gives rise to agroup homomorphism

Extn(B′, A) → Extn+1(B,A)(13.1.11)


dened dually" to (13.1.6); these maps serve as the connecting maps in a long exactsequence

0 // Hom(B,A) // Hom(B′′, A) // Hom(B′, A) //

// Ext1(B,A) // Ext1(B′′, A) // Ext1(B′, A) // · · ·(13.1.12)

naturally associated to (13.1.5). This endows the functors Extn(_ , B) with thestructure of a δ-functor

Ext•(_ , B) : Aop → Ab.(13.1.13)

We see similarly that when A has enough projectives, the Yoneda Ext groupsExtn(A,B) agree with the derived functors of

Hom(_ , B) : Aop → Ab

calculated in the usual way using projective resolutions of objects of A.

13.2. Yoneda pairing. Given three objects A,B,C of A, there is a Yoneda pairing(or Yoneda composition)

Exti(A,B)× Extj(B,C) → Exti+j(A,C)(13.2.1)

obtained by concatenating extensions of the form (13.1.1) in the obvious manner.For example, the connecting morphism" (13.1.6) associated to an exact sequence(13.1.5) is given by Yoneda pairing with the class B ∈ Ext1(B,B′) of that shortexact sequence.

It can be shown that this Yoneda pairing is associative in the evident sense [O].In [BB] it is shown that extensions

E = [0→ B → E → A→ 0] ∈ Ext1(A,B)(13.2.2)G = [0→ C → G→ B → 0] ∈ Ext1(B,C)

have Yoneda pairing

E G = [0→ C → G→ E → A→ 0] ∈ Ext2(A,C)

equal to zero i there is an object F0 of A with a decreasing ltration

0 = F3 ⊆ F2 ⊆ F1 ⊆ F0(13.2.3)

such that the obvious short exact sequences

0→ F1/F2 → F0/F2 → F0/F1 → 0(13.2.4)0→ F2 → F1 → F1/F2 → 0

are isomorphic to E and G, respectively. Little else seems to be known about theYoneda pairing.


13.3. Yoneda Ext is universal. Fix an abelian category A and an object A of A.To ease notation in this section, we will write F • for the δ-functor (13.1.8) given byYoneda Ext in A (13.1):

F • := Ext•(A, _ ) : A → Ab.(13.3.1)

The purpose of this section is to prove:

Theorem 13.3.1. The δ-functor (13.3.1) is universal in the sense of [H, 3.1].

Proof. Given a δ-functor G• : A → Ab and a natural transformation of maps ofabelian categories

α0 : F 0 = Hom(A, _ )→ G0

(a 2-morphism in the 2-category of abelian categories) we must prove that α0 extendsuniquely to a morphism of δ-functors α• : F • → G•. Fix some positive integer nand suppose we have constructed 2-morphisms αi : F i → Gi for i = 1, . . . , n − 1which dene a morphism of (n− 1)-truncated δ-functors" α<n : F<n → G<n. Thismeans that for every exact sequence

0→ B′ → B → B′′ → 0(13.3.2)

in A and every i = 1, . . . , n− 1, the diagram of abelian groups

F i−1B′′δ //

αi−1

F iB′

αi

Gi−1B′′

δ // GiB′

(13.3.3)

commutes, where the horizontal arrows δ are the connecting maps for the δ-functorsF • and G•. It suces to show that there is a unique 2-morphism αn : F n → Gn

such that (13.3.3) commutes for i = n for each exact sequence (13.3.2).

Construction. To construct the 2-morphism αn, we need to construct a grouphomomorphism

αn : F nB = Extn(A,B) → GnB(13.3.4)

natural in B ∈ A. Consider an extension E ∈ YExtn(A,B) as in (13.1.1). Let

K := Im(En → En−1) = Ker(En−1 → En−2).

Set E := En to ease notation. Our exact sequence E determines two exact sequences

K = [0→ B → E → K → 0](13.3.5)Z = [0→ K → En−1 → · · · → E1 → A→ 0].(13.3.6)

To be clear, we should really write K(E), E(E), K(E), and Z(E) to indicate thedependence of K, E, K, and Z on E; this will be important later. By abuse ofnotation, let Z denote the equivalence class of (13.3.6) in Extn−1(A,K) = F n−1K.When n = 1 we have K = A, K = E and we set

Z := Id ∈ Ext0(A,K) = Hom(A,A)


since the previous denition of Z does not really make sense when n = 1. Letαn(E) ∈ GnB be the image of Z ∈ F n−1K under the composition of αn−1 :F n−1K → Gn−1K and the connecting map δ : Gn−1K → GnB associated by theδ-functor G• to the short exact sequence K of (13.3.5).

To complete the proof we need to check the following:

(1) The element αn(E) ∈ GnB in the Construction depends only on the equiv-alence class of E (the image of E ∈ YExtn(A,B) in F nB = Extn(A,B)),hence αn denes a map of sets

αn : F nB = Extn(A,B) → GnB.

(2) The map of sets in (1) is a group homomorphism.(3) The group homomorphism of (2) is natural in B, hence these αn dene a

2-morphism αn : F n → Gn.(4) The diagram (13.3.3) (with i = n) commutes for each exact sequence (13.3.2).(5) The 2-morphism αn is the unique extension of α<n to a morphism of n-

truncated δ-functors α≤n : F≤n → G≤n.

For (1), the relation on YExtn(A,B) given by having the same image under themap of sets

αn : YExtn(A,B) → GnB(13.3.7)

is certainly an equivalence relation, so to show that this equivalence relation containsthe one generated by existence of a morphism," (13.1) it is enough to show thatE,E ′ ∈ YExtn(A,B) have the same image under (13.3.7) when there is a morphismf : E → E ′. LetK, E,K, Z, E be constructed from E as in the Construction and letK ′, E ′, K ′, Z ′, E ′ be the corresponding objects constructed from E ′. The morphismf determines a morphism K → K ′ and hence a morphism F n−1K → F n−1K ′ as in13.1. The key point is that this latter morphism takes Z ∈ F n−1K to Z ′ ∈ F n−1K ′.This is because f gives rise to a commutative diagram with exact rows

0 // K

// En−1 //

· · · // E1

// A // 0

0 // K ′ // E ′n−1 // · · · // E ′1 // A // 0.

Furthermore, f gives rise to a map of short exact sequences K → K ′ (which is theidentity on the left term B of each) and hence to the following solid commutative


diagram of abelian groups:

F n−1K //

''OOOOO

OOF nB

$$Gn−1K //

GnB

F n−1K ′ //

''OOOOO

OOF nB

$$Gn−1K ′ // GnB

(13.3.8)

The horizontal arrows are the connecting morphisms associated by F • and G• tothe short exact sequences K and K ′ and the diagonal arrows are obtained from the2-morphism αn−1. The fact that E and E ′ have the same image under (13.3.7) isclear from the denition of αn, the commutativity of (13.3.8) and the key point."

The top face of the cube in diagram (13.3.8) can also be used to prove (5): Supposeβn extends α<n to a morphism of n-truncated δ-functors F≤n → G≤n. We want toprove βn = αn : F nB → GnB for every B ∈ A. This βn completes (13.3.8) asindicated by the dotted arrows. The key point here is that E ∈ F nB is the image ofZ ∈ F n−1K under the connecting map F n−1K → F nB (this is clear from the recipefor this connecting map in 13.1). It is now immediate from commutativity of thetop face in (13.3.8) that βn(E) = αn(E).

The verication of (2) and (3) is a straightforward exercise with the constructionof αn and can be carried out by the reader as easily as it can be explained by theauthor.

The proof of (4) when n > 1 is rather tautological. Fix some E ∈ F n−1B′′ =Extn−1(A,B′′) represented by an extension

0→ B′′ → En−1 → · · · → E1 → A→ 0.

The image δE of E under the connecting map δ : F n−1B′′ → F nB′ in (13.3.3) isrepresented by the extension

0→ B′ → B → En−1 → · · · → E1 → A→ 0.

But it is clear from the construction of K and Z in the recipe for αn that Z(δE) = Eand K(δE) = B is just the original short exact sequence (13.3.2), so the square(13.3.3) (with i = n) commutes by denition of αn.

The proof of (4) when n = 1 is a little dierent. We want to prove that

F 0B′′ = Hom(A,B′′)δ //

α0

F 1B′ = Ext1(A,B′)

α1

G0B′′

δ // G1B′

(13.3.9)

commutes. Fix some f ∈ Hom(A,B′′). Recall (13.1) that δ(f) ∈ Ext1(A,B′) isthe equivalence class of the short exact sequence on the top row of the commutative


diagram

0 // B′ // E //

A

f

// 0

0 // B′ // B // B′′ // 0

(13.3.10)

where E := B ×B′′ A. By denition, α1(δ(f)) ∈ G1B′ is the image of Id ∈ F 0A =Hom(A,A) under the composition of α0 : F 0A → G0A and the connecting mapG0A→ G1B′ associated by the δ-functor G• to the top row of (13.3.10). The mapof short exact sequences (13.3.10) gives rise to a commutative diagram of abeliangroups

F 0A

%%JJJJJ

J

F 0B′′

G0A //

%%JJJJJ

J G1B′

KKKKKK

KKKKKK

G0B′′ // G1B′.

(13.3.11)

The map F 0A → F 0B′′ in (13.3.11) is the map f∗ : Hom(A,A) → Hom(A,B′′),which takes Id to f . So, if we follow Id ∈ F 0A through the composition

F 0A→ F 0B′′ → G0B′′ → G1B′

in (13.3.11), we get the same result as following f through the composition

F 0B′′ → G0B′′ → G1B′

in (13.3.9). On the other hand, the former is the same thing as following Id throughthe composition

F 0A→ G0A→ G1B′

in (13.3.11), which is the same as the image of f under the composition

F 0B′′ → F 1B′ → G1B′

in (13.3.9).

13.4. Ext comparison. Suppose F : A → B is an exact morphism of abeliancategories. Fix an object A of A. Since F is exact,

Ext•B(FA,F _ ) : A → Ab

is a δ-functor. The functor F induces a natural transformation (2-morphism)

HomA(A, _ ) → HomB(FA,F _ )(13.4.1)

of maps of abelian categories A → Ab. Since Ext•A(A, _ ) is a universal δ-functor(Theorem 13.3.1), the natural transformation (13.4.1) extends uniquely to a mor-phism of δ-functors

Ext•A(A, _ ) → Ext•B(FA, F _ ).(13.4.2)


In fact the maps

ExtnA(A,B) → ExtnB(FA,FB)(13.4.3)

induced by evaluating (13.4.2) at B ∈ A can easily be described explicitly: (13.4.3)simply takes (the equivalence class of) an exact sequence (13.1.1) (viewed as anelement of ExtnA(A,B)) to its image under the exact functor F (viewed as an elementof ExtnB(FA,FB)). Indeed, it is straightforward to check that this recipe for (13.4.3)is well-dened, natural in B, and denes a map of δ-functors (13.4.2) given by theevident map (13.4.1) in degree zero. The maps (13.4.3) dened by this concreterecipe must therefore coincide with the maps constructed by general nonsense bythe uniqueness part of the universality of Ext•A(A, _ ). The following result is clearfrom this description of the maps (13.4.3):

Lemma 13.4.1. Let F : A → B be an exact, fully faithful morphism of abeliancategories. Suppose that A is extension closed in B in the sense that, for anyobjects A′, A′′ ∈ A, and any short exact sequence

0→ FA′ → B → FA′′ → 0

in B, B is in the essential image of F . Then the maps (13.4.3) are isomorphismsfor i = 0, 1.

Let us now assume A and B have enough injectives. We can then give an al-ternative construction of (13.4.3) via spectral sequences. Start with an injectiveresolution I• of B in A, then take an injective Cartan-Eilenberg resolution J•• ofFI• in B. This yields maps

HomA(A, I•)→ HomB(FA, FI•)→ HomB(FA, J••)(13.4.4)

of double complexes of abelian groups, where the two complexes on the left areregarded as double complexes concentrated in degree zero in the vertical" (q")direction. The left map in (13.4.4) is induced by F and the right map is induced byI• → J••.

The nth cohomology of the total complex of HomB(FA, J••) is given by

Hn(TotHomB(FA, J••)) = ExtnB(FA, FB).(13.4.5)

This is because FB → FI• is a quasi-isomorphism, since F is exact, hence thetotal complex of J•• is an injective resolution of FB (note also that applyingHomB(FA, _ ) commutes with forming the total complex of a rst-quadrant doublecomplex). Alternatively, one can compute this cohomology using the spectral se-quence where one rst takes cohomology in the horizontal direction; this degeneratesat E2 = E∞.

The maps of double complexes (13.4.4) induce maps

E → ′E → ′′E(13.4.6)

between the corresponding spectral sequences E, ′E, ′′E, where we take verticalcohomology rst. The second pages (which are the nal pages for E and ′E because


the left two double complexes of (13.4.4) are concentrated in degree zero in thevertical direction) of these spectral sequences are given by:

Ep,q2 = Ep,q

∞ =

ExtpA(A,B), q = 0

0, q = 0(13.4.7)

′Ep,q2 =′ Ep,q

∞ =

Hp(HomB(FA, FI•)), q = 0

0, q = 0(13.4.8)

′′Ep,q2 = Hp(ExtqB(FA,FI•)).(13.4.9)

The nal term ′′Ep,0∞ of the spectral sequence ′′E also maps naturally to the pth

cohomology of the associated double complex (it is the last potentially nontrivialterm in the induced ltration of this cohomology), which we computed in (13.4.5);this yields a map:

′′Ep,0∞ → ExtnB(FA, FB).(13.4.10)

Composing the maps

Ep,0∞ → ′Ep,0

∞ → ′′Ep,0∞(13.4.11)

induced by (13.4.6) and the map (13.4.10), and using the formula (13.4.7) for Ep,0∞ ,

we obtain a map

ExtpA(A,B) → ExtpB(FA, FB).(13.4.12)

One can argue, in the usual way (using universality), that the maps (13.4.12)constructed via spectral sequences above coincide with the maps constructed bygeneral nonsense in the rst paragraph of this section. The basic point is that themaps (13.4.12) constructed above do not depend on the chosen injective resolutionI• or the chosen Cartan-Eilenberg resolution J••.

For later use, we record the following simple fact about the comparison maps(13.4.3):

Lemma 13.4.2. Suppose F : A → B is an exact, fully faithful map of abeliancategories with enough injectives. Suppose B ∈ A admits an injective resolutionB → I• such that each FIp is injective in B. Then (13.4.3) is an isomorphism forall n.

Proof. We can compute (13.4.3) using the spectral sequences recipe above, using aninjective resolution I• of B with the indicated property. Since the FI• are injective,the spectral sequence ′′E above has ′′Ep,q

2 = 0 for q = 0, so ′′Ep,q2 =′′ Ep,q

∞ , the map′Ep,0∞ →′′ Ep,0

∞ is an isomorphism for all p, and the map (13.4.10) is an isomorphismfor all p. Since F is fully faithful, the map Ep,0

∞ → ′Ep,0∞ appearing in (13.4.11) is

also an isomorphism for all p in light of the formulas (13.4.7) for these groups. Themap (13.4.12) is hence a composition of isomorphisms.


13.5. Sheaf Ext. The intention of this section (and the next) is to review the basicnotions of sheaf Ext and to address some subtleties concerning sheaf Ext for quasi-coherent sheaves on bad" (not locally noetherian, say) schemes. These issues arisein the theory of formal smoothness in light of the relationship between Ext groupsand algebra extensions provided by the Fundamental Theorem of the CotangentComplex (Theorem 7.2.1). The standard references for Ext sheaves are [T, 4.2] and[EGA, III0.12.3], though both of those references are fairly simplistic and do notseem to provide a resolution of the issues we will grapple with momentarily.

Let X be a ringed space, F an OX-module. For any OX-module G , the presheaf

U 7→ HomMod(U)(F |U,G |U)(13.5.1)

is again anOX-module (i.e. it is a sheaf), denotedH omX(F ,G ), or justH om(F ,G )if there is no chance of confusion. It is clear from this denition that for any opensubspace U ⊆ X we have

H omX(F ,G )|U = H omU(F |U,G |U).

The functor

H om(F , _ ) : Mod(X) → Mod(X)(13.5.2)G 7→ H om(F ,G )

is left exact and Mod(X) has enough injectives [H, III.2.2], [T, 3.1], so we canconsider the right derived functors

E xti(F , _ ) : Mod(X) → Mod(X)(13.5.3)

G 7→ E xti(F ,G ) := Ri H om(F ,G ).

Lemma 13.5.1. Suppose X is a ringed space and F is an injective OX-module.Then for any open subspace U of X, F |U is an injective OX |U-module.

Proof. This is a formal consequence of the existence of an exact left adjoint i! :Mod(U)→Mod(X) to F 7→ F |U . Recall that, for G ∈Mod(U), the extensionby zero" i!G ∈Mod(X) of G is the sheaf associated to the presheaf

V 7→

G (V ), V ⊆ U

0, otherwise

The stalks of i!G are clearly given by

(i!G )x =

Gx, x ∈ U

0, x /∈ U.

Remark 32. It is tempting to try to repeat the same argument for quasi-coherentsheaves on a scheme X. However, even when G is quasi-coherent, i!G is hardlyevery quasi-coherent. For example, suppose X = A1

k, k a eld, and U ⊆ X isthe complement of the origin 0 ∈ X. I claim i!OU is not quasi-coherent. Sincethe generic point η of X is in U , we have (i!OU)η = OU,η = OX,η = 0 and we


certainly have (i!OU)0 = 0. For any quasi-coherent sheaf F on a scheme X and anypoints x, y of X with x in the closure of y, the generalization map Fx → Fy isalways a localization (the map Fx⊗OX,x

OX,y → Fy is an isomorphism), but this isclearly not the case for the generalization map (i!OU)0 → (i!OU)η, so i!OU can't bequasi-coherent.

Lemma 13.5.2. For a ringed space X and OX-modules F and G , the OX-moduleE xti(F ,G ) is naturally isomorphic to the OX-module associated to the presheaf

U 7→ ExtiOX |U(F |U,G |U),(13.5.4)

where ExtiOX |U is the Ext group for the abelian category of OX |U-modules (thisabelian group has a natural OX(U)-module structure). The restriction maps forthis presheaf are the Ext comparison maps of 13.4 for the exact restriction functorQco(V )→ Qco(U) associated to an inclusion U ⊆ V of open subspaces of X.

Proof. Let F i(F ,G ) denote the sheacation of (13.5.4). For xed F , we haveδ-functors

E xt•(F , _ ), F •(F , _ )Mod(X)→Mod(X)

(one uses exactness of sheacation to see this for the latter). These two δ-functorsagree for i = 0 and the former is clearly eacable, so we obtain the desired naturalisomorphism from general nonsense provided that F •(F , _ ) is eacable, whichfollows from Lemma 13.5.1. Lemma 13.5.3. For a ringed space X, OX-modules F and G , i ∈ N, and an opensubspace U ⊆ X, we have a natural isomorphism

E xtiX(F ,G )|U = E xtiU(F |U,G |U).(13.5.5)

Proof. This is immediate from Lemma 13.5.2 and the fact that sheacation com-mutes with restricting a presheaf to an open subspace. Alternatively, one uses theGrothendieck spectral sequences relating the derived functors for the two composi-tions in the commutative" diagram:

Mod(X)H om(F ,_ )

//

_ |U

Mod(X)

_ |U

Mod(U)H om(F |U,_ )

// Mod(U)

Since one of the two functors in each such composition is the exact functor _ |U ,these spectral sequences degenerate to yield the desired isomorphism. In order tomake use of the Grothendieck spectral sequence, one needs to know that _ |U takesinjective OX-modules to H omU(F |U, _ )-acyclic OU -modules, which is clear fromLemma 13.5.1. Remark 33. One can make the same denitions for complexes of OX-modulesF , G because the abelian category of complexes of OX-modules also has enoughinjectives (Theorem 13.7.5). Lemma 13.5.2 remains true (by the same proof), where


the Ext on the right side of (13.5.4) denotes the Ext group in the abelian categoryof complexes of OX |U -modules (13.8). Lemma 13.5.3 remains true in this moregeneral setting.

Lemma 13.5.4. Let X be a ringed space, F a locally free OX-module of locallynite rank. Then E xti(F ,G ) = 0 for every i > 0 and every OX-module G .

Proof. Since we can check that a sheaf is zero locally, (13.5.5) reduces us to provingthe lemma when F = On

X for some integer n ≥ 0. But then H om(OnX , _ ) is exact

since it is identied with the functor

G 7→⊕n

G =∏n

G ,

so its higher derived functors vanish. Lemma 13.5.5. Let A → B be a at ring homomorphism. For any M,N ∈Mod(A), there is a natural isomorphism

ExtiB(M ⊗A B,N ⊗A B) = ExtiA(M,N)⊗A B

of B-modules for every i. For M ∈Mod(A) and N ∈Mod(B), there is a naturalisomorphism

ExtiB(M ⊗A B,N) = ExtiA(M,N)⊗A B

of A-modules for every i.

Proof. For the rst statement, take a projective resolution P• →M of the A-moduleM . Extension of scalars always takes projectives to projectives because it has anexact right adjoint given by restriction of scalars. Since A → B is at, P• ⊗A B ishence a projective resolution of M ⊗A B. Then we compute

Exti(M ⊗A B,N ⊗A B) = Hi(HomB(P• ⊗A B,N ⊗A B))

= Hi(HomA(P•, N ⊗A B))

= Hi(HomA(P•, N)⊗A B)

= Hi(HomA(P•, N))⊗A B

= ExtiA(M,N)⊗A B

using atness of A→ B to commute the tensor product with taking homology in thepenultimate equality. The second statement of the lemma is proved similarly.

Let A be a ring, M,N ∈Mod(A), X := SpecA. Suppressing the equivalence ofcategories

Mod(A) = Qco(X),

the Ext comparison" map (13.4.3) of 13.4 for the exact, fully faithful inclusionfunctor Qco(X) →Mod(X), takes the form of a natural A-module map

ExtnA(M,N) → ExtnMod(X)(M∼, N∼).(13.5.6)


Theorem 13.5.6. Let A be a ring, M,N ∈ Mod(A), X := SpecA. There is anatural map of OX-modules

ExtnA(M,N)∼ → E xtn(M∼, N∼)(13.5.7)

which is an isomorphism under any of the following hypotheses:

(1) n = 0 or n = 1.(2) A is noetherian.(3) Locally on X, there exists an exact sequence of OX-modules

Pn−2 → · · · → P0 →M∼ → 0

with each Pi free of nite rank.

Furthermore, if A is noetherian andM and N are nitely generated, then ExtnA(M,N)is nitely generated for every n ∈ N, so

E xtn(M∼, N∼) ∼= ExtnA(M,N)∼

is coherent.

Proof. To construct the natural map in question, it suces, by basic sheaf theory andthe description of E xtn(M∼, N∼) as a sheacation in Lemma 13.5.2, to constructmaps

ExtnA(M,N)∼(U) → ExtnMod(U)(M∼|U,N∼|U)

as U runs over a basis for the open subsets of X which are natural for inclusions ofopens U ⊆ V in the chosen basis. We take as our chosen basis for X the set of aneopens of the form U = SpecAf as f runs over elements of A. Using the standardformula for the sections (resp. restriction) of a quasi-coherent sheaf over (resp. to)such a basic open, our maps should take the form

ExtnA(M,N)⊗A Af → ExtnMod(U)((M ⊗A Af )∼, (N ⊗A Af )

∼).

Since the localization A → Af is at, the left hand side is naturally identiedwith ExtnAf

(M ⊗A Af , N ⊗A Af ) and the map we want to construct is just thenatural map (13.5.6) mentioned above, for the ring Af . It is straightforward to seethat these maps are compatible with restriction, thus the construction of the maps(13.5.7) is complete. Since the maps we just constructed on basic opens are clearlyisomorphisms, the n = 0 statement in (1) is immediate. The n = 1 statement in (1)is similarly an isomorphism by Lemma 13.4.1 because an extension of quasi-coherentsheaves is again quasi-coherent [H, II.5.7].

For (2) we use the (rather nontrivial!) fact that we can nd an injective resolutionN → I• such that each (Ip)∼ is an injective object in Mod(X) (not merely inQco(X) = Mod(A))this is proved in [RD] in the section where they discuss theclassication of injectives on a locally noetherian scheme. Both sides of (13.5.7) arethen the nth cohomology of

HomA(M, I•)∼ = HomA(M∼, (I•)∼)

(using the n = 0 isomorphism).


In light of the n = 0, 1 result, there is nothing to prove in (3) unless n ≥ 2, sowe can assume n ≥ 2 and, by induction, that the result is known for smaller n (forall A, M , N). Since we can check isomorphy of (13.5.7) locally on ane opens ofX and we have the formula (13.5.5), we can assume the exact sequence in questionexists globally on X, so Pi = F∼i for some nitely generated free A-modules Fi. LetK be the kernel of F0 →M so we have exact sequences

0→ K → F0 →M → 0(13.5.8)Pn−2 → · · · → P1 → K∼ → 0.(13.5.9)

From (13.5.8) we obtain a commutative diagram with exact rows

0 // Extn−1A (K,N)∼

// ExtnA(M,N)∼

// 0

0 // E xtn−1(K∼, N∼) // E xtn(M∼, N∼) // 0

where the top row is obtained from the long exact ExtA(_ , N) sequence and thevanishings Extn−1(F0, N) = 0 and Extn(F0, N) = 0 (because F0 is a projective A-module), the bottom row is obtained from the long exact E xt(_ , N∼) sequence andthe vanishings E xtn−1(P0, N

∼) = 0 and E xtn(P0, N∼) = 0 of Lemma 13.5.4, and

the vertical arrows are the natural maps in question. The left vertical arrow is anisomorphism by induction because we have the exact sequence (13.5.9), hence theright vertical arrow is an isomorphism.

The furthermore" is proved by the same induction argument (since one can alwaysnd (globally) the sort of exact sequence in (3)) and does not require the use of theresult (2). One uses only the simple fact that HomA(M,N) is a nitely generatedA-module when M and N are nitely generated modules over a noetherian ringA. Corollary 13.5.7. Let X be a scheme, F , G quasi-coherent OX-modules. ThenH om(F ,G ) and E xt1(F ,G ) are quasi-coherent OX-modules. If X is locally noe-therian and F and G are quasi-coherent (resp. coherent), then E xtn(F ,G ) is quasi-coherent (resp. coherent) for every n ∈ N.

Proof. The is immediate from the theorem and the formula (13.5.5) applied to aneopen subschemes U of X. Remark 34. In [EGA, I.9.1], the fact that H om(F ,G ) is quasi-coherent when Fand G are quasi-coherent is proved only under the superuous assumption that Fis of locally nite presentation.

Lemma 13.5.8. A nitely generated module M over a noetherian ring A is projec-tive i Ext1A(M,N) = 0 for each nitely generated A-module N .

Proof. When M is nitely generated and A is noetherian, we can certainly nd anexact sequence of A-modules

0→ N → An →M → 0(13.5.10)


for some n ∈ N and some nitely generated A-module N . When Ext1A(M,N) = 0,the map

Hom(An, N)→ Hom(N,N)

induced by the inclusion N → An is surjective, so we can lift the identity map ofN to some r ∈ Hom(An, N). This r : An → N is a retract of N → An, hence itprovides a splitting of (13.5.10), so M is a direct summand of the free A-module An

and is hence projective. Theorem 13.5.9. Let X be a scheme, F a quasi-coherent OX-module. The fol-lowing are equivalent:

(1) For any ane open subspace U = SpecA of X, F |U is isomorphic to M∼

for a projective A-module M .(2) For every open subspace U of X and every quasi-coherent sheaf G on U , we

have E xt1U(F |U,G ) = 0.(3) There exists an ane open cover Ui = SpecAi of X, projective Ai-modules

Mi and isomorphisms F |Ui∼= M∼

i for each i.

If X is quasi-separated (which holds, for example, if the space underlying X is locallynoetherian), then these conditions are also equivalent to

(4) For every quasi-coherent sheaf G on X, we have E xt1X(F ,G ) = 0.

If X is locally noetherian and F is coherent, these conditions are also equivalent to

(5) For every coherent sheaf G on X, we have E xt1X(F ,G ) = 0.

Proof. For (1) implies (2), we can check that E xt1U(F |U,G ) = 0 locally on U , sowe can assume U = SpecA is ane. By hypothesis, we can write F |U = M∼ for aprojective A-module M (and G = N∼ for some A-module N). Then we compute

E xt1(F |U,G ) = Ext1A(M,N)∼

= 0

using the isomorphism of Theorem 13.5.6(1) for the rst equality and projectivityof M for the second.

For (2) implies (1), write F |U ∼= M∼ for some A-module M . We need to showM is projective. For this it suces to show Ext1A(M,N) = 0 for every A-moduleN , for then Hom(M, _ ) will be exact. But Ext1A(M,N)∼ ∼= E xt1(F |U,N∼) byTheorem 13.5.6(1), which vanishes by hypothesis, so Ext1A(M,N) = 0.

Obviously (1) implies (3).

For (3) implies (1), choose ane covers Vij = SpecAij of each U ∩Ui, so F |Vij =(F |Ui)|Vij

∼= (Mi ⊗AiAij)

∼. Note that extension of scalars _ ⊗AiAij takes pro-

jectives to projectives. We thus reduce to the case where X = U = SpecA. WriteF ∼= M∼ for some A-module M ; we want to show M is projective. It is equivalentto show that every short exact sequence

0→ N → E →M → 0


of A-modules splits. The hypothesis in (3) and what we have already proved impliesthat this sequence splits locally. But the obstruction to gluing local splittings toa global one lies in H1(X,H om(M∼, N∼)), which vanishes since X is ane andH om(M∼, N∼) = Hom(M,N)∼ is quasi-coherent.

Obviously (2) implies (4), even without the quasi-separated hypothesis. For theconverse under that hypothesis, suppose U ⊆ X is an arbitrary open subschemeand G ∈ Qco(U). We want to prove E xt1(F |U,G ) = 0. It suces to prove thatE xt1(F |U,G ) restricts to zero on each ane open subspace V ⊆ U . Since X isquasi-separated, the inclusion map i : V → X is quasi-compact (and separaratedsince it is monic) so the pushforward i∗ takes quasi-coherent sheaves on V to quasi-coherent sheaves on X ([H, 5.8(c)] or [EGA, I.9.2.2]). We then compute

E xt1(F |U,G )|V = E xt1(F |V,G |V )

= E xt1(F |V, (i∗(G |V ))|V )

= E xt1(F , i∗(G |V ))|V= 0

using Lemma 13.5.3, the fact that (i∗(G |V ))|V = G |V , Lemma 13.5.3 again, andthe hypothesis of (4) for the respective equalities.

Assume now that X is locally noetherian. Obviously (4) implies (5). It remainsto prove that (5) implies, say, (3) (when F is coherent). Take any ane coverUi = SpecAi of X where the Ai are noetherian rings. Since F is coherent, wecan write F |Ui = M∼

i for some nitely generated Ai-module Mi. To show that thisMi is projective, it suces, by Lemma lem:nitelygeneratedprojective, to prove thatExt1Ai

(Mi, N) = 0 for each nitely generated Ai-module N . By standard results onextension of coherent sheaves [EGA, I.9.4.8], we can nd a coherent sheaf G on Xsuch that G |Ui = N∼. Now we compute

0 = E xt1X(F ,G )|Ui

= E xt1Ui(F |Ui,G |Ui)

= E xt1Ui(M∼

i , N∼)

= Ext1Ai(Mi, N)∼

using the hypothesis in (5), the restriction formula of Lemma 13.5.3, and the n = 1isomorphism of Theorem 13.5.6(1), so Ext1(Mi, N) = 0. Denition 27. A quasi-coherent sheaf F on a scheme X satisfying the equivalentproperties of Theorem 13.5.9 is called Qco locally projective.

13.6. Quasi-coherent sheaf Ext. The theory of sheaf Ext, as introduced in theprevious section, has some drawbacks when working on a scheme X which is notlocally noetherian. For example, even if F , G are quasi-coherent, we (we" meansthe author") cannot prove, in general, that E xtn(F ,G ) is quasi-coherent for n > 1.Indeed, we cannot prove this even when X is ane, since we do not know whetherthe map (13.5.7) of Theorem 13.5.6 is an isomorphism in complete generality (we


suspect it is not). In this section, we will discuss a slight variant of the theory ofsheaf Ext which can be used to remedy this defect. The results of this section areused only very slightly elsewhere in the text (namely in the proof of (4) implies (5)"in Theorem 7.5.1) and will be of no interest to the reader who assumes all schemesare locally noetherian. We rst recall the following fact:

Theorem 13.6.1. (Enochs-Estrada) For any scheme X, the abelian categoryQco(X) satises the axiom AB 5) of [T, 1.5] and has a generator [T, 1.9], henceQco(X) has enough injectives by Grothendieck's general criterion [T, 1.10.1].

Proof. See [EE].

We can now attempt to remedy the aw mentioned above by considering, forF ∈ Qco(X), the right derived functors of the functor

H omX(F , _ ) : Qco(X) → Qco(X)(13.6.1)

(recall from Corollary 13.5.7 that H omX(F ,G ) is quasi-coherent when F and Gare quasi-coherent). Note that we can only speak of Rn H omX(F ,G ) when G isquasi-coherent, but we always know this is a quasi-coherent sheaf (this is already anadvantage over the usual sheaf Ext of the previous section). However, this theoryof sheaf Ext has a defect of its own: we do not know whether it is compatible withrestriction to open subspaces in the sense that we do not know whether

(Rn H omX(F ,G ))|U = Rn H omU(F |U,G |U)(13.6.2)

for U an open subscheme of X.

Instead, for F ,G ∈ Qco(X), we let E xtnQco(X)(F ,G ) denote the sheaf associatedto the presheaf

U 7→ ExtnQco(U)(F |U,G |U).(13.6.3)

Note that the Ext here is calculated in Qco(U) as opposed to Mod(U).

We will see momentarily (Lemma 13.6.4) that the OX-modules E xtnQco(X)(F ,G )are quasi-coherent and (Corollary 13.6.3), in the case of an ane scheme X, theycoincide with the right derived functors of (13.6.1) mentioned above.

The usual Ext comparison" maps

ExtnQco(U)(F |U,G |U) → ExtnMod(U)(F |U,G |U)

associated to the exact inclusions Qco(U) →Mod(U) (13.4) are natural in U andhence dene a morphism of presheaves

ExtnQco(_ )(F |_ ,G |_ ) → ExtnMod(_ )(F |_ ,G |_ ).(13.6.4)

The sheacation of the presheaf on the right is E xtn(F ,G ) (Lemma 13.5.2), sothe sheacation of (13.6.4) denes a map of OX-modules

E xtnQco(X)(F ,G ) → E xtn(F ,G ).(13.6.5)


Remark 35. We do not really need the enough injectives" result of Theorem 13.6.1to dene the sheaves E xtnQco(X)(F ,G ). This is because we can dene a sheaf bysheafying a presheaf dened only on the open subsets in some basis for the topologyof a space X. For a scheme X, the ane opens U = SpecA of X form a basis forits topology, and Qco(U) = Mod(A) certainly has enough injectives.

The next lemma shows that quasi-coherent sheaf Ext on an ane scheme isbetter-behaved" than the usual sheaf Ext (c.f. Theorem 13.5.6).

Lemma 13.6.2. Let A be a ring, M,N ∈Mod(A), X = SpecA. For every n ∈ Nthere is a natural isomorphism

ExtnA(M,N)∼ = E xtnQco(X)(M∼, N∼)

of OX-modules. In particular, E xtnQco(X)(M∼, N∼) is quasi-coherent.

Proof. By general sheaf theory and the description of E xtnQco(X)(M∼, N∼) as the

sheaf associated to the presheaf (13.6.3), it is enough to construct an isomorphismof OX(U)-modules

ExtnA(M,N)∼(U) = ExtnQco(U)(M∼|U,N∼|U)

for each U in a basis U of open subspaces of X which is compatible with the restric-tion maps for an inclusion U ⊆ V of opens in our basis U . We of course use thebasis U consisting of ane opens U = SpecAf , f ∈ A. The desired isomorphism isthen the composition of the sequence of natural isomorphisms

ExtnA(M,N)∼(U) = ExtnA(M,N)⊗A Af

= ExtnAf(M ⊗A Af , N ⊗A Af )

= ExtnQco(U)(M∼|U,N∼|U).

The rst isomorphism is the usual formula for sections of a quasi-coherent sheaf on abasic open U = SpecAf , the second isomorphism is from Lemma 13.5.5 (the localiza-tion A→ Af is certainly at), and the nal isomorphism is from the equivalence ofcategoriesMod(Af ) = Qco(U) and the standard formulaM∼|U = (M⊗AAf )

∼.

Remark 36. The map considered in Theorem 13.5.6 is nothing but the compositionof the isomorphism of Lemma 13.6.4 and the comparison" map (13.6.5) (with F =M∼, G = N∼). It follows from Theorem 13.5.6(2) that (13.6.5) is an isomorphismwhenever X is locally noetherian, since this can be checked on ane opens.

Corollary 13.6.3. Let A be a ring, M an A-module, X = SpecA. The quasi-coherent sheaf Ext functors

E xt•Qco(X) : Qco(X) → Qco(X)

are the right derived functors of

H om(M∼, _ ) : Qco(X) → Qco(X).


Proof. This is a restatement of the lemma using the fact that the equivalence ofcategories Mod(A) = Qco(X) identies the functor

H omX(M∼, _ ) : Qco(X) → Qco(X)

with the functor

HomA(M, _ ) : Mod(A) → Mod(A).

Lemma 13.6.4. Let X be a scheme, F ,G ∈ Qco(X). For any open subschemeU ⊆ X and any n ∈ N, we have

E xtnQco(X)(F ,G )|U = E xtnQco(U)(F |U,G |U).

If U = SpecA is ane and we choose isomorphisms F ∼= M∼, G ∼= N∼ forM,N ∈Mod(A), then there is an induced isomorphism

E xtnQco(X)(F ,G )|U = ExtnA(M,N)∼.

In particular, the OX-module E xtnQco(X)(F ,G ) is quasi-coherent.

Proof. The rst statement follows from the fact that sheacation commutes withrestricting to an open subset. The second statement is then immediate from therst statement, plus Lemma 13.6.2. For the nal statement, note that the propertyof being quasi-coherent is local.

We now give a criterion for a quasi-coherent sheaf to be Qco locally projectivewhich is perhaps a bit less contrived than the criteria in Theorem 13.5.9.

Theorem 13.6.5. Let X be a scheme. A quasi-coherent OX-module F is Qcolocally projective (i.e. the equivalent conditions of Theorem 13.5.9 hold) i, for everyopen subspace U of X, every quasi-coherent sheaf G on U , and every n > 0, we have

E xtnQco(U)(F |U,G ) = 0.

Proof. This follows easily from the lemma and the criteria of Theorem 13.5.9 for Fto be Qco locally projective.

In the remainder of the section, we work out an abstract characterization of quasi-coherent sheaf Ext along the lines of the universal δ-functor" characterization ofother derived functors.

Denition 28. Let X be a scheme. A functor F : Qco(X) → Qco(X) is calledmonotone i, for every F ∈ Qco(X) and every open subspace U ⊆ X with F |U = 0we have (FF )|U = 0. Similarly, a δ-functor F • : Qco(X) → Qco(X) is calledmonotone i each F n is monotone in the previous sense.

Lemma 13.6.6. Let X be a quasi-separated scheme, F • : Qco(X) → Qco(X) aδ-functor, i : U → X the inclusion of an ane open subscheme. Then

i∗ : Mod(U) → Mod(X)


takes Qco(U) into Qco(X) and there is a natural morphism

i∗F •_ → i∗F •i∗i∗_(13.6.6)

of δ-functors Qco(X)→ Qco(U) which is an isomorphism if F • is monotone.

Proof. The rst statement follows from standard algebraic geometry ([H, 5.8(c)] or[EGA, I.9.2.2]) since the hypothesis that X is quasi-separated implies i is quasi-compact (certainly i is separated since it is monic). The natural morphism is con-structed by applying F • to the adjunction morphism Id → i∗i

∗, then applying theexact functor i∗. To see that

i∗F •i∗i∗F → i∗F •F

is an isomorphism when F • is monotone, note that the kernel K and cokernel Cof the adjunction map F → i∗i

∗F restrict to 0 on U because the adjunction maprestricts to an isomorphism on U . Let F → I be the image of the adjunctionmorphism so we have a short exact sequence

0→ K → F → I → 0.

Applying i∗ to the long exact sequence associated by F • to this short exact sequenceand using i∗F •K = 0 (since i∗K = 0 and F • is monotone), we nd that i∗F •F →i∗F •I is an isomorphism. Arguing similarly with the short exact sequence

0→ I → i∗i∗F → C → 0,

we nd that i∗F •I → i∗F •i∗i∗F is an isomorphism. The composition of the two

aforementioned isomorphisms is the desired isomorphism. Theorem 13.6.7. Let X be a scheme, F a quasi-coherent sheaf on X. Then

E xt•Qco(X)(F , _ ) : Qco(X) → Qco(X)

is a monotone δ-functor. In fact, if X is quasi-separated, it is universal amongmonotone δ-functors G• : Qco(X)→ Qco(X).

Proof. The fact that E xt•Qco(X)(F , _ ) is a δ-functor is clear from exactness ofsheacation and the fact that

Ext•Qco(U)(F |U, _ ) : Qco(U)→Mod(OX(U))

is certainly a δ-functor for each open subscheme U ⊆ X. This δ-functor is monotonein light of the compatibility with restriction to open subschemes in Lemma 13.6.4.

For the nal statement, set F • := E xt•Qco(X)(F , _ ) to ease notation. SupposeG• : Qco(X) → Qco(X) is a monotone δ-functor and α0 : F 0 → G0 is a 2-morphism. We must prove that α0 extends uniquely to a morphism of δ-functorsα• : F • → G•. Let i : U → X be the inclusion of an ane open subscheme of X, sowe have an (exact) map of abelian categories

i∗ : Qco(U) → Qco(X)

(rst statement of Lemma 13.6.6).


The δ-functors F • and G• give rise to δ-functors

F •U , G•U : Qco(U) → Qco(U)

dened by F •U _ := i∗(F •i∗_ ) (and similarly with F replaced by G) and α0 givesrise to a 2-morphism α0

U : F 0U → G0

U . The restriction formula in Lemma 13.6.4implies that

F •U _ = i∗ E xtnQco(X)(F , i∗_ )

= E xt•Qco(U)(i∗F , i∗i∗_ )

= E xt•Qco(U)(F |U, _ ).

Since U is ane, we know from Corollary 13.6.3 that F •U is universal, so α0U extends

uniquely to a morphism of δ-functors α•U : F •U → G•U . Explicitly, this α•U is amorphism of δ-functors

α•U : i∗(F •i∗_ ) → i∗(G•i∗_ ).

Precomposing with i∗ we get a morphism of δ-functors

α•U : i∗(F •i∗i∗_ ) → i∗(G•i∗i

∗_ ).

Rewriting this using the natural isomorphisms of Lemma 13.6.6 (since F • and G•

are monotone) and writing |U instead of i∗, we get a morphism of δ-functors

α•U : (F •_ )|U → (G•_ )|U.

These morphisms will glue to yield the desired unique extension of α0. Indeed,to see that these agree on overlaps U ∩ V , we need only cover U ∩ V with openanes W and use the uniqueness part of the universality of F •W to conclude that(α•U)|W = α•W = (α•V )|W . The uniqueness of the resulting α• is proved similarly,since we can check that any two lifts agree on an ane cover.

Corollary 13.6.8. Let X be a quasi-separated scheme, F a quasi-coherent sheafon X. If the δ-functor given by the right derived functors of

H om(F , _ ) : Qco(X) → Qco(X)

is monotone (which holds, for example, if it satises the compatibility (13.6.2) withrestriction to open subschemes, or if every quasi-coherent sheaf on X admits amonomorphism into an injective quasi-coherent sheaf with the same set-theoreticsupport), then it coincides with the δ-functor E xt•Qco(X)(F , _ ) of Theorem 13.6.7.

13.7. Categories of complexes. We consider a xed abelian category A through-out this section. Let Ch(A) denote the category of chain complexes in A, whoseobjects L consist of objects Li of A (one for each i ∈ Z), and A-morphismsdi : Li → Li−1, called boundary maps, such that di−1di = 0 for each i ∈ Z. Amorphism f : L→ M in Ch(A) consists of A-morphisms fi : Li → Mi which com-mute with the boundary maps (i.e. difi = fi−1di for all i ∈ Z). Given an object L


of Ch(A) and an i ∈ Z, we let

Zi(L) := Ker(di : Li → Li−1)

Bi(L) := Im(di+1 : Li+1 → Li)

Hi(L) := Zi(L)/Bi(L)

denote the cycles, boundaries, and homology of L. These dene functors

Zi, Bi, Hi : Ch(A) → A.(13.7.1)

An object L of Ch(A) is called acyclic i Hi(L) = 0 for all i ∈ Z.The category Ch(A) is also an abelian category, with kernels and cokernels formed

degreewise." Consequently, a sequence of complexes

0→ L′ → L→ L′′ → 0

is exact in B i0→ L′n → Ln → L′′n → 0

is exact in A for each n ∈ N. We will often consider the full abelian subcategoriesCh≥0(A) and Ch[0,∞)(A) of Ch(A) whose objects consist, respectively, of thoseL ∈ Ch(A) with Li = 0 for i < 0 (respectively with Li = 0 for i < 0 and Li = 0 forall suciently large i).

Remark 37. Of course there are many other bounded" versions of Ch(A). Manyresults in this section have variants for these other categories of bounded complexesthat can be easily worked out by the reader. The category Ch≥0(A) is of the greatestimportance in the rest of the text.

Lemma 13.7.1. Let B denote one of the abelian categories Ch(A), Ch≥0(A), orCh[0,∞)(A). Then each object of B admits a monomorphism into an acyclic objectof B. Variant: For each object L of Ch(A) (resp. Ch≥0(A), Ch[0,∞)(A)) there isan object N of Ch(A) (resp. Ch≥0(A), Ch[0,∞)(A)) with Hi(N) = 0 for all i (resp.for i > 0, for i > 0) and an epimorphism f : N → L.

Proof. Given L ∈ B, let M ∈ B be the complex with Mi = Li ⊕ Li−1 and withboundary maps

ei :=

(0 Id

0 0

): Li ⊕ Li−1 → Li−1 ⊕ Li−2.

(We write ei for the boundary maps for M to distinguish them from the boundarymaps di for L.) Clearly ei−1ei = 0 and M is acyclic since

Zi(M) = Bi(M) = Li ⊕ 0 ⊆Mi

for each i. The map f : L→M with

fi :=

(Id

di

): Li → Li ⊕ Li−1


is clearly monic and is a well-dened B-morphism because

eifi =

(0 Id

0 0

)(Id

di

)

=

(di0

)

=

(di

di−1di

)

=

(Id

di−1

)di

= fi−1di.

The Variant is proved similarly by taking Ni := Li ⊕ Li+1 (except one must takeNi := 0 for i = −1 in the two respective" cases, else N−1 might be non-zero) andfi := (Id, di+1). Corollary 13.7.2. Let B denote one of the abelian categories Ch(A), Ch≥0(A), orCh[0,∞)(A). Any injective object of B is acyclic. Variant: Any projective object Pof Ch(A) (resp. Ch≥0(A), Ch[0,∞)(A)) is acyclic (resp. has Hi(P ) = 0 for i = 0).

Proof. Say I is an injective object of B. By the lemma, there is a monomorphismI →M with M acyclic; since I is injective, this monomorphism is the inclusion of asummand: M = I ⊕ J , so I is a direct summand of an acyclic complex and is henceacyclic since formation of homology commutes with formation of the direct sum oftwo complexes. Lemma 13.7.3. The homology functor H0 : Ch≥0(A) → A is right exact. Thefunctors H• : Ch≥0(A) → A form an eacable, hence universal, δ-functor (i.e. Hn

is the nth left derived functor of H0).

Proof. The rst statement and the fact that H• is a δ-functor follow from the SnakeLemma. This δ-functor is eacable by the Variant of Lemma 13.7.1. Lemma 13.7.4. Let B denote one of the abelian categories Ch(A), Ch≥0(A), orCh[0,∞)(A). For any integer i, the (exact) functor B→ A given by L 7→ Li has anexact right adjoint A→ B taking A to the two-term complex [A→ A] sitting in de-grees i+1 and i with di+1 = Id. This right adjoint takes injectives to injectives. Thefunctor L 7→ Li also has an exact left adjoint which takes projectives to projectives.

Proof. The adjunction bijection

HomB(L, [A→ A]) → HomA(Li, A)

takes f : L → [A → A] to fi : Li → A and its inverse takes an A-morphismg : Li → A to the B-morphism L → [A → A] given by g in degree i and fdi+1 indegree i+ 1.


The left adjoint to L→ Li takes A ∈ A to the two-term complex [A→ A] sittingin degrees i and i−1 with di = Id. When B is Ch≥0(A) or Ch[0,∞)(A), a variation isneeded when i = 0: the left adjoint simply takes A ∈ A to A, regarded as a complexconcentrated in degree zero. The adjunction bijection is similar to the one above.

It is clear from these descriptions that these adjoints are exact. The statementabout injectives to injectives" is a formal consequence of the existence of an exactleft adjoint, and similarly for the projectives to projectives".

Theorem 13.7.5. Let B denote one of the abelian categories Ch(A), Ch≥0(A),Ch[0,∞)(A). If A satises any of the properties listed below, the same property isalso satised by B:

(1) Any of the usual abelian category axioms AB 3), AB 4), AB 5), AB 6) (andtheir duals") as in [T, 1.5].

(2) Existence of enough injectives.(3) Existence of enough projectives.(4) Existence of a set of generators [T, 1.9].(5) Existence of a set of cogenerators.

Proof. For (1) it is enough to note that limits in B can be formed degreewise [T,1.6.1], [T, 1.7(e)].

For (2), suppose A has enough injectives and let L be an object of B. Choosea monomorphism fi : Li → Ii in A, with Ii injective in A, for each i ∈ Z. ByLemma 13.7.4, the complex M(i) = [Ii → Ii] concentrated in degrees i + 1 and iis injective and the map fi corresponds to a B-morphism g(i) : L → M(i) withg(i)i : Li → M(i)i = Ii equal to fi, hence injective. The product of the g(i) yieldsa map g : L → M :=

∏iM(i). Note that this product exists (in B) and is equal

to the corresponding direct sum (without any assumptions on A) because the setof complexes M(i) : i ∈ Z is degree-wise nite (sums and products can be formeddegree-wise). We have Mi = Ii ⊕ Ii−1. For similar reasons, gi = (fi, fi−1di) : Li →Mi = Ii ⊕ Ii−1 is monic, hence g is monic and M is injective since it is a product ofinjectives. The proof of (3) is similar.

For (4), suppose Gj is a set of generators for A. Let Hi,j ∈ B denote the imageof Gj under the left adjoint to L 7→ Li (Lemma 13.7.4). I claim Hi,j is a set ofgenerators for B. Indeed, suppose, L ( M is a nontrivial subobject of an object Mof B. Then there must be some i ∈ Z such that Li ⊆ Mi is a nontrivial subobjectof the object Mi of A, so there is an index j and a map w : Gj → Mi that doesn'tfactor through Li. Then the image of w under the bottom adjunction isomorphismin the commutative diagram

HomA(Gj, Li) _

HomB(Hi,j, L) _

HomA(Gj,Mi) HomB(Hi,j,M)


doesn't factor through L. The proof of (5) is similar. Corollary 13.7.6. Let A be an abelian category with enough injectives. An object Lof Ch(A) (resp. Ch≥0(A), Ch[0,∞)(A)) is injective i L is acyclic, each Li is injectivein A, and each Zi(L) = Bi(L) is injective in A. Equivalently, L is injective i thereare injective objects Ii of A for i ∈ Z (resp. i ≥ 0, resp. i ≥ 0 with Li = 0 forsuciently large i) such that L is isomorphic to the complex M with Mi = Ii⊕ Ii−1and boundary map

di =

(0 Id

0 0

).

Proof. The equivalently" is justied by noting that, when Zi(L) is injective, theshort exact sequence

0→ Zi(L)→ Li → Bi−1(L)→ 0

splits. We saw in the proof of (2) from the previous theorem that any L admitsa monomorphism into a complex as described in the present theorem. When L isinjective, this monomorphism is the inclusion of a direct summand, and it is clearthat a direct summand of a complex as in the statement of the theorem is also ofthe same type.

13.8. Hyper Ext. In an eort to keep things as low-tech" as possible in thesenotes (that is, to avoid any serious discussion of derived categories), we will nowformulate the so-called hyper Ext" groups in the usual language of derived functorsof half exact functors between abelian categories.

In this section we consider a xed abelian category A. To ease notation, we letB := Ch≥0(A) denote the abelian category of chain complexes in A supported innitely many non-negative degrees, as studied in 13.7.

Fix a complex L ∈ B. As in any abelian category, the functor

HomB(L, _ ) : B → Ab

is left exact. Assuming A has enough injectives, the abelian category B also hasenough injectives (Theorem 13.7.5) and we can form the right derived functors

Extn(L, _ ) := Rn HomB(L, _ ),

which we will call hyper Ext groups. Even if B does not have enough injectives, wecan still dene Extn(L, _ ) using Yoneda Ext (13.1) to obtain a universal δ-functor

Ext•(L, _ ) : B → Ab

(Theorem 13.3.1). The functor HomB(L, _ ) is contravariantly functorial in L,hence so are these hyper Ext groups, and by general nonsense one also has

Rn HomB(_ ,M) = Extn(_ ,M)

where we now think of

HomB(_ ,M) : Bop → Ab


as a left exact functor. A short exact sequence

0→ L′ → L→ L′′ → 0

in B naturally gives rise to a long exact sequence of abelian groups:

0 // HomB(L′′,M) // HomB(L,M) // HomB(L

′,M) //

// Ext1(L′′,M) // Ext1(L,M) // Ext1(L′,M) // · · ·(13.8.1)

Remark 38. If A is the category of A-modules for a ring12 A, then every abeliangroup" in question also has a natural A-module structure, so one can replace Ab"everywhere with A."

We can view A as the full subcategory of B consisting of complexes supported indegree zero. If M ∈ A ⊆ B, then for any L ∈ B one has an isomorphism of abeliangroups

HomB(L,M) = HomA(H0(L),M)

natural in L (i.e. H0 is left adjoint to the inclusion A ⊆ B) and hence a commutativediagram of left exact functors:

BopHomB(_ ,M)

//

Hop0

Ab

AopHomA(_ ,M)

66mmmmmmmmmmmmmm

Assuming A (hence B) has enough injectives, the Grothendieck Spectral Sequencerelating the right derived functors of the functors involved13 is a rst quadrantspectral sequence

Ep,q2 = Rp HomA(R

q Hop0 _ ,M) =⇒ Rp+q HomB(_ ,M).

Written in a less ridiciulous way, we have a spectral sequence

Ep,q2 = ExtpA(Hq(L),M) =⇒ Extp+q(L,M)(13.8.2)

contravariantly functorial in L ∈ B and covariantly functorial in M ∈ A ⊆ B. Inparticular, there is a short exact sequence of low-order terms"

0→ Ext1A(H0(L),M)→ Ext1(L,M)→ Hom(H1(L),M)→ Ext2A(H0(L),M)(13.8.3)

contravariantly functorial in L ∈ B and covariantly functorial in M ∈ A ⊆ B.

Remark 39. One can construct the exact sequence (13.8.3) explicitly without usingthe Grothendieck Spectral Sequence and without using the assumption that A hasenough injectives. It is useful to have an explicit description of each of the mapsin (13.8.3). The leftmost map in (13.8.3) is the usual map induced by the aug-mentation" map a : L→ H0(L) and the usual functoriality of Ext groups (the map

12All of our rings" are commutative. If A isn't commutative, then HomMod(A)(M,N) only hasthe structure of a module over the center of A.13To check that one has such a spectral sequence one must rst check that if I ∈ B is injective,then H0(I) ∈ A is injective.


a is viewed as a B-morphism, viewing H0(L) ∈ A as an object of B by the usualinclusion A ⊆ B). The next map Ext1(L,M) → Hom(H1(L),M) in (13.8.3) canbe described as follows: Viewing Ext1(L,M) as the group of isomorphism classes ofshort exact sequences

0→M → E → L→ 0

in B (13.1), the map in question takes (the isomorphism class) of such an exactsequence to the connecting map H1(L) → M = H0(L) in the resulting long exactsequence of homology groups. The map HomA(H1(L),M) → Ext2A(H0(L),M) in(13.8.3) can be described as follows: An A-morphism f : H1(L) → M can becomposed with the natural surjection

Z1(L) := Ker(d1 : L1 → L0) → H1(L)

to yield a map Z1(L) → M . The image of f under the map in question is theelement of the Yoneda Ext group Ext2A(H0(L),M) obtained by pushing out" theexact sequence

0→ Z1(L)→ L1 → L0 → H0(L)→ 0(13.8.4)

alongZ1(L)→M (i.e. the image of the class of (13.8.4) under the map Ext2A(H0(L),Z1(L))→Ext2A(H0(L),M) induced by Z1(L) → M). With these explicit descriptions of themaps involved one can (laboriously) check the exactness of (13.8.3) by hand."

Lemma 13.8.1. Suppose f : L′ → L is a quasi-isomorphism in B (a map such thatHn(f) : Hn(L

′)→ Hn(L) is an isomorphism for all n ∈ N). Then

f ∗ : Extn(L,M) → Extn(L′,M)

is an isomorphism for any M ∈ A ⊆ B and any n ∈ N.

Proof. When f is a quasi-isomorphism, the map induced by f on the E2-terms ofthe spectral sequences (13.8.2) for L and L′ is an isomorphism, hence f induces anisomorphism as indicated on the abutments of these spectral sequences.

Lemma 13.8.2. Suppose L ∈ B is cohomologically formal (quasi-isomorphic to acomplex with all boundary maps zero). Then for any M ∈ A ⊆ B and any p ∈ Nwe have

Extp(L,M) =

p⊕n=0

Extp−nA (Hn(L),M).

Proof. In light of the previous lemma, we can assume all the boundary maps in Lare zero. Then L = ⊕n Hn(L)[n] in B, where Hn(L)[n] denotes the object Hn(L)of A viewed as an object of B (complex) supported in degree n. Note that thisdirect sum is nite since we assume the complexes in B are bounded. Since derivedfunctors commute with (nite) direct sums, we have

Extp(L,M) = ⊕n Extp(Hn(L)[n],M).(13.8.5)


On the other hand, if A is any object of A and n ∈ N, the spectral sequence (13.8.2)(applied to L = A[n]) degnerates at E2 to yield

Extp(A[n],M) = Extp−nA (A,M).(13.8.6)

Putting together (13.8.5) and (13.8.6) yields the result. Remark 40. Any complex with nonzero cohomology in at most one degree is coho-mologically formal. Indeed, suppose L has cohomology in at most degree n. Thenthe maps of complexes given by the vertical arrows below are quasi-isomorphisms:

· · · // Ln+1

dn+1 // Lndn // Ln−1 // · · ·

· · · // Ln+1

=

OO

dn+1 // Ker dn

//

OO

0

OO

// · · ·

· · · // 0 // Hn(L) // 0 // · · ·

We will often apply Lemma 13.8.2 in the form below.

Lemma 13.8.3. Suppose L ∈ B has Hn(L) = 0 for n = 0 (resp. n = 1). Then

Extn(L,M) = ExtnA(H0(L),M) (resp. Extn−1A (H1(L),M) )

for all n ∈ N and all M ∈ A ⊆ B.

We will be particularly interested in the groups Ext1(L,M) for L ∈ B, M ∈A ⊆ B. As in any abelian category, one has a Yoneda" description of the Extgroups in terms of extensions [Y]. In particular, Ext1(L,M) is in natural bijectivecorrespondence with the set of isomorphism classes of short exact sequences

0→M → E → L→ 0(13.8.7)

in B, where an isomorphism is a commutative diagram

0 // M // E ′

// L // 0

0 // M // E // L // 0

in B. The complex M is supported in degree zero, so, up to isomorphism, a shortexact sequence of complexes (13.8.7) is nothing but a commutative diagram in A ofthe form

L1

Id // L1

0 // M // E0

// L0// 0

(13.8.8)

where the row is exact. (Exactness of (13.8.7) implies En → Ln is an isomorphismfor n > 0, so up to isomorphism of extensions we can assume En = Ln for n > 0.)


Lemma 13.8.4. Consider an exact diagram in A as below.

L′1

// L1//

L′′1 //

0

0 // L′0 // L0// L′′0 // 0

Regarding the columns as two-term complexes in B (in degrees 1, 0), we have anexact sequence

H1(L′) // H1(L) // H1(L

′′) //

// H0(L′) // H0(L) // H0(L

′′) // 0

in A. For any M ∈ A ⊆ B, we have an exact sequence of abelian groups

0 // HomB(L′′,M) // HomB(L,M) // HomB(L

′,M) //

/ / Ext1(L′′,M) // Ext1(L,M) // Ext1(L′,M)

Proof. The rst statement is the Snake Lemma. For the second statement, let

K1 := Ker(L1 → L′′1) = L′1/Ker(L′1 → L1)

and K0 := L′0 so we have a commutative diagram

L′1

// K1

L′0 K0

in A where L′1 → K1 is surjective. Since the kernel of L′1 → L1 maps to L′0 via thezero map (because L′0 → L0) is injective, the vertical maps in the above diagramhave the same image. This diagram can hence be viewed as a map of complexesL′ → K inducing an isomorphism on H0 and a surjection on H1. By construction ofthe complex K = [K1 → K0], we have a short exact sequence of complexes

0→ K → L→ L′′ → 0

and hence a long exact sequence

0 // HomB(L′′,M) // HomB(L,M) // HomB(K,M) //

// Ext1(L′′,M) // Ext1(L,M) // Ext1(K,M)

of abelian groups. Comparing this exact sequence with the one we want, we reduceto proving that the natural map

HomB(K,M) → HomB(L′,M)

is an isomorphism and the natural map

Ext1(K,M) → Ext1(L′,M)


is injective. The rst map is an isomorphism because H0(L′) → H0(K) is an iso-

morphism. For the injectivity statement, we consider the exact diagram

0 // Ext1A(H0(L′),M)

// Ext1(L′,M)

// HomA(H1(L′),M)

0 // Ext1A(H0(K),M) // Ext1(K,M) // HomA(H1(K),M)

relating the low-order terms in the natural spectral sequences (13.8.2) for K and L′.The left vertical map is an isomorphism because H0(L

′)→ H0(K) is an isomorphism,and the right vertical map is injective because H1(L

′)→ H1(K) is surjective, so themiddle vertical arrow is injective by a diagram chase.

Remark 41. The exact sequences in the above lemma are natural in the inputdiagram because the constructions of 1) the new complex K and 2) the map L′ → Kin the proof are natural.

Lemma 13.8.5. Let f : L → L′ be a B-morphism inducing isomorphisms Hi(f) :Hi(L) ∼= Hi(L

′) for i = 0, 1. Then the induced map

f ∗ : Exti(L′,M) → Exti(L,M)

is an isomorphism for i = 0, 1 for any M ∈ A ⊆ B.

Proof. The i = 0 statement is clear since we have a natural isomorphism

Ext0(L,M) = HomA(H0(L),M)

for all L ∈ B, M ∈ A ⊆ B. For the i = 1 statement, we need only apply the FiveLemma in the exact diagram

0 // Ext1A(H0(L′),M)

∼=

// Ext1(L′,M)

f∗

// HomA(H1(L′),M)

∼=

// Ext2A(H0(L′),M)

∼=

0 // Ext1A(H0(L),M) // Ext1(L,M) // HomA(H1(L),M) // Ext2A(H0(L),M))

relating the low-order terms in the natural spectral sequences (13.8.2) for L andL′the indicated maps are isomorphisms under our assumptions.

14. Groupoid Fibrations

This section contains some miscellaneous category-theoretic aspects of the theoryof stacks : categories bered in groupoids, 2-commutative diagrams, 2-bered products,and liftings, formal smoothness / étaleness, representability, etc. Nothing here isstrictly necessary elsewhere in the notes, but the reader who wants to understandthe proper setup for various moduli problems might nd something useful in thissection.


Denition 29. Let C → D be a functor. Denote the image of an object or mor-phism of C under this functor by underlining it. A C morphism f : c′ → c is calledcartesian (relative to the functor C → D) i, for any C morphism g : c′′ → c,the map h 7→ h yields a bijection from completions of the left diagram below tocompletions of the right diagram below.

c′′h //

g@

@@@@

@@@ c′

f

c

c′′ //

g>

>>>>

>>>

c′

f

c

The functor C→ D is called a category bered in groupoids (over D) (or a groupoidbration) i every C morphism is cartesian.

Denition 30. Let C → D be a functor and let d be an object of D. The bercategory over d, denoted F−1(d) or Cd, is the category whose objects are thoseobjects c of C with c = d and whose morphisms are the C morphisms f : c → c′

with f = Idd. If F is a groupoid bration, then one sees easily that each bercategory is a groupoid (a category all of whose morphisms are isomorphisms).

14.1. Examples. The terminology is probably motivated by the following example.To a topological space X we can associate a category I(X) whose objects are pointsx ∈ X and where a morphism from x0 to x1 is a homotopy class [γ] of maps oftriples

γ : (I, 0, 1)→ (X, x0, x1)

where I = [0, 1] is the unit interval. You really have to use homotopy classes ofmaps so that you can dene the composition of morphisms. The category I(X)is covariantly functorial in X: A continuous map f : X → Y induces a functorI(f) : I(X) → I(Y ) dened on objects by I(f)(x) := f(x) and on morphisms byI(f)([γ]) := [fγ]. The functor I(f) is a groupoid bration i the map f has theso-called path lifting property. In particular, if f is a Serre bration, then I(f)is a groupoid bration. The category I(X) should be viewed as a sort of coarsetruncation of the simplicial set of maps from the standard cosimplicial topologicalspace ∆• to X.

The simplest example of a groupoid bration is the identity functor Id : D→ D.The next simplest example is the groupoid bration D/d → D associated to anobject d of D. Objects of D/d are D morphisms f ′ : d′ → d and a morphism from(f ′′ : d′′ → d) to (f ′ : d′ → d) is a D morphism g : d′′ → d′ such that f ′g = f ′′. Thefunctor given by (f ′ : d′ → d) 7→ d′ is clearly a groupoid bration.

Slightly more generally, any contravariant functor X : Dop → Sets (i.e. anypresheaf on D) determines a groupoid bration over D (often denoted by the sameletter X ) whose objects are pairs (d, x) where d is an object of D and x ∈ X (d).A morphism (d′, x′)→ (d, x) in this category is a D morphism f : d′ → d such thatX (f)(x) = x′. The functor (d, x) 7→ d is a groupoid bration.


The primary example of groupoid brations is that of global quotients". SupposeX : Dop → Sets is a presheaf on D and G : Dop → Groups is a functor (presheafof groups) with an action of G on X . This means: For each object d ∈ D, we havean action of the group G (d) on the set X (d) and for each D-morphism f : d′ → d,the restriction map X (f) : X (d) → X (d′) is G (f) : G (d) → G (d′) equivariant.Then we can construct a category [X /G ], called the quotient of X by G as follows.Objects of [X /G ] are pairs (d, x) where d is an object of D and x ∈ X (d). An[X /G ] morphism from (d′, x′) to (d, x) is a pair (f, u) where f : d′ → d is a Cmorphism and u is an element of the group G (d′) such that u · x′ = X (f)(x).Notice that the objects of [X /G ] are the same as the objects of X , but there aremore morphisms in [X /G ]. There is a functor [X /G ] → D dened on objects by(d, x) 7→ d and on morphisms by (f, u) 7→ f . It is easy to check that this functor isa groupoid bration.

Notice that the category [X /G ] is a rened version of the quotient X /G taken inpresheaves: Instead of just saying that two objects in the same orbit become equal,we actually keep track of the element of the group taking one to the other. Now, ifG acted freely (i.e. with trivial stabilizers), then we wouldn't need a name for theelement identifying two points of the same orbit because this would be unambiguous.In this sense, the quotient [X /G ] is always a free quotient," though it comes atthe cost of working in a fancier categorical framework.

14.2. 2-categories.

Denition 31. A 2-category C consists of the following data:

(1) A set obC (often abusively denoted C) called the set of objects of C.(2) A set HomC(X, Y ) of 1-morphisms for each ordered pair (X,Y ) of objects

of C.(3) A composition function

HomC(X,Y )× HomC(Y, Z) → HomC(X,Z)

(f, g) 7→ gf

for each ordered triple (X, Y, Z) of objects of C.(4) A set Hom2

C(f, g) of 2-morphisms for each ordered pair (X, Y ) of objects ofC and each ordered pair (f, g) of elements of HomC(X,Y ).

(5) A vertical composition function

HomC(f, g)× HomC(g, h) → HomC(f, h)

(α, β) 7→ βα

dened for each ordered pair (X,Y ) of objects of C and each ordered triple(f, g, h) of elements of HomC(X,Y ).

(6) For each ordered triple of objects (X,Y, Z), the following two types of hori-zontal composition function: A function

_ ∗ f : Hom2C(g, h) → Hom2

C(gf, hf)

α 7→ α ∗ f


dened for each ordered pair (g, h) of elements of HomC(Y, Z), and eachelement f ∈ HomC(X, Y ), as well as a function

k ∗ _ : Hom2C(i, j) → Hom2

C(ki, kj)

β 7→ k ∗ β

dened for each ordered pair of elements (i, j) of elements of HomC(X, Y )and each element k ∈ HomC(Y, Z).

These data are required to satisfy the following conditions:

(1) The set obC in (1) with the morphisms dened in (2) and compositiondened in (3) is a category.

(2) For each xed ordered pair (X, Y ) of objects ofC, the set of objectsHomC(X,Y )with morphisms given by the 2-morphisms in (4) and composition of suchmorphisms given by the vertical composition in (5) forms a category.

(3) For each ordered triple (X,Y, Z) of objects of C and each f ∈ HomC(X, Y )the function given on objects by g 7→ gf and on morphisms by α 7→ α ∗ f(the rst horizontal composition operation in (6)) denes a functor

f ∗ : HomC(Y, Z) → HomC(X,Z)

when these are viewed as categories as in (2). Similarly, for each k ∈HomC(Y, Z), the map given on objects by i 7→ ki and on morphisms byβ 7→ k ∗ β (the second horizontal composition operation in (6)) denes afunctor

k∗ : HomC(X, Y ) → HomC(X,Z).

(4) We have (α∗f)∗g = α∗ (fg), i∗ (j ∗β) = (ij)∗β, and (f ∗α)∗g = f ∗ (α∗g)whenever these operations are dened.

(5) For each triple (X,Y, Z) of objects of C, each ordered pair (g, h) of elementsof HomC(X,Y ), each ordered pair (k, l) of elements of HomC(Y, Z), eachelement α ∈ Hom2

C(g, h), and each element β ∈ Hom2C(k, l), the diagram

kgk∗α //

β∗g

kh

β∗h

lgl∗α // lh

commutes in the category HomC(kg, lh).

14.3. Groupoid brations as a 2-category. Fix a base category" D. We willoften consider a commutative diagram of functors as below.

XF //

!!BBB

BBBB

B Y

~~

D

(14.3.1)


Here we really mean commutative" in the strictest possible sense of an actual equal-ity between the two functors X → D obtained from this diagram. (Later on (14.4)we will have a more lax notion of commutativity.) We will denote both of the func-tors X → D and Y → D by underlining as in Denition 29, so the commutativitymeans Fx = x for all objects and morphisms x of X . In this situation, the functorF induces a functor between the ber categories Fd : Xd → Y for each object d ofD.

Functors toD form a 2-category denotedCat/D. An object ofCat/D is a functorX → D (usually denoted by underlining as in Denition 29). We often suppressthe structure functor X → D in the notation. A 1-morphism in Cat/D from Xto Y is a commutative diagram of functors (14.3.1). Given two one morphismsF,G : X ⇒ Y , a 2-morphism η : F → G is a natural transformation of functorssuch that ηx = Id for every object x of X . The same denitions make groupoidbrations to D into a full 2-subcategory CFG/D of Cat/D whose objects are thoseobjects X of Cat/D where the structure map X → D is a groupoid bration.If η : F → G is a 2-morphism in CFG/D, then the condition ηx = Id that ηxis a morphism in a ber category of Y implies that ηx is an isomorphism, henceη is an invertible 2-morphism. All 2-morphisms in CFG/D are invertible. Whenspeaking about CFG/D, we will usually say morphism" and homotopy" insteadof 1-morphism" and 2-morphism".

Denition 32. We say that two 1-morphisms G,H in a 2-category are homotopici there exists an invertible 2-morphism η : G → H (called a homotopy from G toH). A 1-morphism G is called an equivalence i there is a 1-morphism H in theopposite direction (called an inverse) such that both GH and HG are homotopicto the identity. We say that a 1-morphism R : Y → X is a deformation retractof a 1-morphism I : X → Y i RI = Id and there is an invertible 2-morphismη : IR→ IdY which restricts to the identity on X ". (In a general 2-category, thismeans the 2-morphism η ∗ I : IRI = I → I is the identity. In Cat/D, this meansthat η(I(x)) = Id for each object x of X .)

For example, in the global construction quotient of 14.1, we have a CFG/D1-morphism X → [X /G ] dened on objects by the identity and on morphims byf 7→ (f, 1).

Lemma 14.3.1. Let F be a morphism in Cat/D as in (14.3.1), if F is full (resp.faithful, fully faithful, an equivalence in Cat/D), then Fd is full (resp. ...) for everyobject d of D. If F is a morphism in CFG/D, then each converse also holds.

Proof. The notation C(c′, c) := HomC(c′, c) for a category C is convenient. For

d ∈ D and x, x′ ∈Xd, we have a commutative diagram of sets

Xd(x′, x)

Fd //

Yd(Fx′, Fx)

X (x′, x)

F // Y (Fx′, Fx)


where the vertical arrows are the inclusions. But the very fact that (14.3.1) com-mutes says that in fact this diagram is cartesian; the rst three statements followsimmediately. Suppose F is an equivalence with inverse G, so there are homotopiesη : GF → IdX and ζ : FG→ IdY . The denition of a ((a)n invertible) 2-morphismin Cat/D ensures that ηx : GFx → x is morphism in the ber category Xx whichis invertible as an X -morphism, which implies that its inverse is also a morphismin the ber category Xx, so for any d ∈ D, η denes an invertible natural transfor-mation

η|Xd : (GF )|Xd → Id

to the identity functor of Xd, and similarly for ζ, so Fd is an equivalence of categorieswith inverse Gd.

For the rest of the proof we suppose X → D and Y → D are groupoid brations.Suppose each Fd is faithful and let us prove that F : X (x′, x) → Y (Fx′, Fx) isinjective for arbitrary objects x, x′ ∈ X . Suppose Ff = Fg. Then in particularf = g, so since X → D is a groupoid bration there is a unique h : x′ → x′

with h = Id and hf = g. Applying F we have FhFf = Fg and Ff = f = Id,but of course we also have Fh Id = Fg so since Y → D is a groupoid bration,we must have Ff = Id. But Fx′ is faithful, so f = Id, so f = g. Now supposeeach Fd is full and let use prove that F : X (x′, x) → Y (Fx′, Fx) is surjective forarbitrary objects x, x′ ∈ X . Consider an arbitrary Y -morphism g : Fx′ → Fx.Since X → D is a groupoid bration there is an X -morphism f : x′ → x suchthat f = g. Then we have two Y -morphisms Ff, g : Fx′ → Fx lying over the sameD-morphism, so since Y → D is a groupoid bration there is h : Fx′ → Fx′ suchthat Ffh = g. Since Fx′ is full, there is k : x′ → x′ with k = Id and Fk = h. Butthen F (fk) = FfFk = Ffh = g.

It remains only to prove that F is an equivalence when each Fd is an equiva-lence. This is a variant of the usual argument equivalence equals fully faithfulplus essentially surjective." For each object d ∈ D, pick an inverse Gd : Yd → Xd

for Fd : Xd → Yd and homotopy ηd : FdGd → Id. Dene a CFG/D morphismG : Y → X as follows. For each object y ∈ Y , set Gy := Gyy. For a Y mor-phism f : y′ → y, let Gf : Gy′ → Gy be the unique X -morphism such that FGfcompletes the diagram

FGy′

ηy′ (y′)

// y′

f

FGy

ηy(y)// y

(14.3.2)

(there is a unique such morphism because there is certainly a unique completionof the solid diagram in Y since the horizontal arrows are isomorphisms, and thatunique completion in Y is the image under F of a unique X -morphism because weknow from what we've already proved that F is fully faithful). From the uniquenessof this completion, one sees that G denes a functor Y → X which commutes


with the functors to D because the horizontal arrows in (14.3.2) lie over identitymaps in D. Similarly, the natural transformation η : FG → Id dened on y ∈ Yby η(y) := ηy(y) denes a homotopy FG → Id. Furthermore, if the morphismf : y′ → y happens to be a morphism in a ber category (i.e., y′ = y =: d, f = Idd),then

FGy′

FGdf

ηd(y′)

// y′

f

FGy

ηd(y) // y

commutes because ηd is a natural transformation, hence we have Gf = Gdf , so ournewly constructed functor G agrees with the old functors Gd on each ber category,so each Gd is also an equivalence and we can construct a homotopy ζ : GF → Id bythe same procedure.

Lemma 14.3.2. If F : C→ D is a faithful functor then the group homomorphism

F : AutC(c) → AutD(Fc)(14.3.3)

is injective for every c ∈ C. If C is a groupoid, the converse holds. Similarly, if Fis full and C is a groupoid, then (14.3.3) is surjective for every c ∈ C.

Proof. The rst statement is trivial since AutC(c) is a subset of HomC(c, c).

Suppose (14.3.3) is injective and we want to show F is faithful. We need to showthat

F : HomC(c, c′) → HomD(Fc, Fc′)

is injective for all c, c′ ∈ C. If HomC(c, c′) is empty, there is nothing to do, so we can

assume there is some C-morphism g : c→ c′. But C is a groupoid, so all morphismsfrom c to c′ are isomorphisms, and we have a commutative diagram

HomC(c, c′)

F

f 7→g−1f // AutC(c, c)

F

HomD(Fc, Fc′)f 7→Fg−1f // HomD(Fc, Fc)

where the top horizontal arrow is bijective and the right vertical arrow is injectiveby our assumption on F .

Suppose F is full and C is a groupoid. Then for any f ∈ AutD(Fc) we can nda g ∈ HomC(c, c) such that g = Ff . But C is a groupoid, so g ∈ AutC(c). Thisproves that (14.3.3) is surjective.

Remark 42. Even if C, D are groupoids and (14.3.3) is bijective for every c ∈ C,you cannot conclude that F is full. Take C to be the groupoid with two objectsc, c′ whose only maps are the identity maps and take D to be the groupoid withtwo objects Fc, Fc′, but where there is a unique map between any two objects. Let


F : C → D be the inclusion. All automorphism groups in C and D are trivial, so(14.3.3) is trivially bijective, but

F : HomC(c, c′) → HomD(Fc, Fc′)

is the map from the empty set to the one-element set, which is not surjective, so Fisn't full.

14.4. 2-commutative diagrams. This whole section is basically just a bunch ofdenitions. For a while we will work with a general 2-category C. We will denoteobjects of C using capitol Roman letters W,X, Y, Z, etc., 1-morphisms of C usinglower-case Roman letters f, g, etc., and 2-morphisms using lower-case Greek let-ters. We usually say morphism" (resp. homotopy") instead of 1-morphism" (resp.invertible 2-morphism").

Denition 33. A 2-commutative diagram (or homotopy commutative) in C is apair

S

i

a // X

f

Tb // Y

η : fa→ bi(14.4.1)

consisting of a diagram of C-morphisms as on the left and a homotopy η betweenthe two compositions, as indicated.

Given a 2-commutative diagram (14.4.1), we dene the groupoid of liftings in(14.4.1), denoted Lift((14.4.1)) (or just Lift if (14.4.1) is clear from context) asfollows. An object of Lift, called a lift, is a triple (l, α, β) consisting of aC-morphisml : T → X, and homotopies α : a → li, β : fl → b which are compatible with thehomotopy η in the sense that

faη //

f∗α AAA

AAAA

Abi

fliβ∗i

??~~~~~~~~

(14.4.2)

commutes. A morphism in Lift from (l′, α′, β′) to (l, α, β) is a homotopy γ : l′ → lmaking the obvious diagrams"

fl′f∗γ //

β′?

????

???

fl

β

b

and aα′

~~~~~~~~~

α

???

????

l′iγ∗i

// li

(14.4.3)

commute. Note that all morphisms in Lift are isomorphisms, so Lift is a groupoid.

Denition 34. We say that the 2-commutative diagram (14.4.1) has a lift (resp.has an essentially unique lift, has a unique lift, has at most one lift, really has atmost one lift) i the groupoid of liftings Lift dened above is not the empty category


(resp. has exactly one object up to isomorphism, has one object up to isomorphismand that object has no nontrivial automorphisms, has at most one object up toisomorphism, has at most one object up to isomorphism and if it has an object thenthat object has no nontrivial automorphisms).

Denition 35. Let I be a set of morphisms in C. Let I-b (resp. -ét, -sét, -rad,-srad) denote the class of C-morphisms f : X → Y such that any 2-commutativediagram (14.4.1) has a lift (resp. has an essentially unique lift, has a unique lift, hasat most one lift, really has at most one lift).

Denition 36. Suppose Esp is a category of spaces (e.g. the category of ringedspaces, locally ringed spaces, schemes, schemes over a xed base scheme, analyticspaces, etc.) and C = CFG/Esp. Let I be the set" of all square zero closedembeddings of spaces, viewed as a class of morphisms in C by taking images un-der the Yoneda functor Esp → CFG/Esp. Dene the class of formally smooth(resp. formally étale, strongly formally étale, formally unramied, strongly formallyunramied) C-morphisms to be I-b (resp. -ét, -sét, -rad, -srad). It follows fromLemma 14.4.1 below that for representable morphisms in C, there is no dierencebetween formally étale" and strongly formally étale" and there is no dierencebetween formally unramied" and strongly formally unramied".

Lemma 14.4.1. If C = Cat/D is the 2-category of categories over a xed basecategory D, and the functor f in the 2-commutative diagram (14.4.1) is faithful,then the automorphism group of every object of Lift is trivial.

Proof. Suppose γ : l → l is an automorphism of a lift (l, α, β) and t is an arbitraryobject of t. Then evaluating the diagram of invertible natural transformations onthe left of (14.4.3) on t yields a commutative diagram

f(l(t))f(γ(t))

//

β(t) ##GGG

GGGG

Gf(l(t))

β(t)wwwwwwww

b(t)

of isomorphisms in the category Y , which obviously requires f(γ(t)) = Id. Since fis faithful this implies γ(t) = Id.

Suppose we have another 2-commutative diagram:

X

f

g // X ′

f ′

Y

h // Y ′

ζ : f ′g → hf(14.4.4)


Then we can put it on the right" of (14.4.1) to obtain a new 2-commutative diagram

S

i

ga // X ′

f ′

Y

hb // Y ′

θ := (h ∗ η)(ζ ∗ a) : f ′ga→ hbi(14.4.5)

called the horizontal composition of (14.4.1) and (14.4.4). We dene a functor

Lift((14.4.1)) → Lift((14.4.4))

as follows. On objects, we map (l, α, β) to (gl, g ∗ α, (h ∗ β)(ζ ∗ l)). To check thatthis is well-dened, we need to check that the big outer triangle" in the diagram

f ′ga

θ

%%ζ∗a //

f ′∗(g∗α)

&&

hfa

h∗(f∗α)

h∗η // hbi

hfli

h∗(β∗i)

==

f ′gli

((h∗β)(ζ∗l))∗i

PP

ζ∗(li)

OO

commutes. This holds because the top triangle commutes (denition of θ), the leftsquare commutes (use the penultimate 2-category axiom to see that f ′ ∗ (g ∗ α) =(f ′g) ∗ α and h ∗ (f ∗ α) = (hf) ∗ α, then use the nal 2-category axiom to see that

(f ′g)aζ∗a //

(f ′g)∗α

(hf)a

(hf)∗α

(f ′g)(li)ζ∗(li)

// (hf)(li)

commutes), the upper triangle on the right commutes (it is the image of the com-mutativity condition (14.4.2) satised by (l, α, β) under the functor h∗), and thelower right triangle commutes (use the penultimate 2-category axiom and the axiomabout functoriality of i∗). On morphisms, we map γ : (l′, α′, β′)→ (l, α, β) to

g ∗ γ : (gl′, g ∗ α′, (h ∗ β′)(ζ ∗ l′)) → (gl, g ∗ α, (h ∗ β)(ζ ∗ l)).(14.4.6)


To check that (14.4.6) is well-dened we rst check that the diagram below com-mutes.

f ′gl′f ′∗(g∗γ)

//

ζ∗l′

f ′gl

ζ∗l

hfl′

h∗β′!!D

DDDD

DDD

h∗(f∗γ)// hfl′

h∗β

hb

The triangle commutes because it is the image of the left diagram in (14.4.3) underthe functor h∗ and the square commutes because we rst nd f ′ ∗ (g ∗ γ) = (f ′g) ∗ γand h∗ (f ∗γ) = (hf)∗γ using the penultimate 2-category axiom, then we nd that

(f ′g)l′(f ′g)∗γ

//

ζ∗l′

(f ′g)l

ζ∗l

(hf)l′(hf)∗γ

// // (hf)l

commutes by the last 2-category axiom. The last commutativity we need to checkis just the image of the right diagram in (14.4.3) under g∗.

There is a similarly-dened functor

Lift((14.4.4)) → Lift((14.4.5))(14.4.7)

given on objects by

(l, α, β) 7→ (lb, (l ∗ η)(α ∗ a), β ∗ h)

and on morphisms by γ 7→ γ ∗ b. (Here l : Y → X ′ is a 1-morphism and α : g → lf ,β : f ′l→ h are homotopies compatible with ζ.)

14.5. 2-bered products. Suppose (F : C → D) is a groupoid bration over Dand

Gi : (Fi : Ci → D)→ (F : C→ D) (i = 1, 2)

are 1-morphisms in CFG/D. That is, we have a (strictly) commutative diagram ofcategories as below.

C1G1 //

F1 AAA

AAAA

AC

F

C2G2oo

F2~~

D

The bered product C1×CC2 of the Gi is the following category: Objects are triples(c1, c2, u) where ci is an object of ci with F1(c1) = F2(c2) and u : G1(c1) → G2(c2)is a C-morphism in the ber category of C over the object F1(c1) = F2(c2) of D. A


Morphism in C1 ×C C2 from (c′1, c′2, u′) to (c1, c2, u) is a pair (f1, f2) consisting of

Ci morphisms fi : c′i → ci making the diagram

G1(c1)G1(f1)//

u

// G1(c′1)

u′

G2(c2)

G2(f2)// G2(c′2)

(14.5.1)

commute in C.

There is an obvious diagram of groupoid brations over D as below.

C1 ×C C2π2 //

π1

C2

G2

C1

G1 // C

This diagram does not commute, but there is a tautological homotopy η : G1π1 →G2π2 dened by η(c1, c2, u) := u. This diagram is characterized up to an appropriatenotion of equivalence by a universal property we leave to the reader to work out andcheck.

Exercises

Exercise 1. Let X = (X,OX) be a ringed space, I ⊆ OX an ideal (ideal sheaf").Suppose that: 1) X is locally quasi-compact, 2) I is stalkwise nilpotent, and 3)locally on X, I is generated by a nite subset of its global sections. Conclude thatI is locally nilpotent. Conclude that for locally nitely generated ideal sheaves onschemes, locally nilpotent" and stalkwise nilpotent" are equivalent. (If you havetrouble with the rst statement, just prove this last statement.)

Exercise 2. Let A → B be a ring homomorphism, M an A-module, N a B-module (sometimes regarded as an A-module by restriction of scalars). Supposethat TorA1 (M,B) = 0 and TorB1 (M ⊗A B,N) = 0. Prove, in the most elementaryway you can, that TorA1 (M,N) = 0. We used this in the proof of Proposition 2.3.1.

Exercise 3. Show that the bered product of two maps of schemes fi : Xi → Y(i = 1, 2) coincides with the bered product taken in ringed spaces when at leastone of the fi is a closed embedding. That is, in a cartesian diagram of schemes

Xp2 //

p1

X2

f2

X1f1 // Y

where, say, f1 is a closed embedding show that the underlying diagram of topologicalspaces is also cartesian and

OX = p−11 OX1 ⊗f−1OYp−12 OX2


where f := f1p1 = f2p2. If I ⊆ OY is the ideal of the closed embedding f1 : X1 → Y ,show that p2 : X → X2 is the closed subscheme of X2 whose ideal J is the inverseimage of I under f2 (the ideal of OX2 generated by the image of f ♯

2 : f−12 I → OX2 ,or equivalently, the image of the map f ∗2 I → OX2 of quasi-coherent OX2-modules).

Exercise 4. Check that the two maps in the proof of Theorem 3.3.1 are inverse andthat they are B-linear.

Exercise 5. Fill in the missing exactness checks in the proof of Theorem 3.7.1.

Exercise 6. Let A be an abelian category. Show that a diagram

A→ B → C → 0

in A where the composition A→ C is zero is exact i the diagram of abelian groups

0→ HomA(C,D)→ HomA(B,D)→ HomA(A,D)

is exact for every object D of A.

Exercise 7. Show that a formally smooth map of sheaves of rings is stalkwiseformally smooth. (Hint: Use the right adjoint x∗ to the stalk functor x−1.)

Exercise 8. Suppose A→ B is a surjection of rings whose kernel I satises I = I2.Prove that A→ B is formally étale. Construct an example of such an A→ B.

Exercise 9. Let A be a sheaf of rings on a space X. Show that an A-module M isat i Mx is a at Ax-module for every x ∈ X. (Hint: Show that the right adjointx∗ : Mod(Ax)→Mod(A) to the stalk functor is exact and (x∗N)x = N .)

Exercise 10. Let k be a eld, A an artinian k-algebra. Show that A is formallysmooth over k i A is a nite product of elds, each of which is formally smooth overk. (A criterion for a eld extension to be formal smooth is given in Theorem 5.7.4,but that result is irrelevant for this exercise.) Hint: Use the fact that every artinianring is a nite product of local artinian rings to reduce to the case where (A,m) isa local artinian k-algebra. The key point now is to show that if (A,m) is formallysmooth over k, then (A,m) must be reduced (hence A must be a eld by basicstructure theory of artinian rings). For this, show that you can nd a non-zerom ∈ m such that mm = 0. Use this m to construct a square zero thickening wherethere is no lift... you can take the nilpotent thickening to be K[x]/x3 → K[x]/x2

where K is the residue eld of A.

Exercise 11. Let X be a scheme whose underlying topological space is noetherianand all of whose stalks OX,x are integral domains. Show that X is a nite disjointunion of integral schemes. (Recall that a scheme is called integral i its structuresheaf is a sheaf of integral domains.)

Remark 43. There are rings A (necessarily not noetherian) such that Ap is anintegral domain for every p ∈ SpecA, SpecA is connected, and A is not an integraldomain.

Exercise 12. Let B → A be a surjection of rings whose kernel contains b.


(1) Let P be a projective B module and let d : P → Ker b be a surjective Bmodule morphism. Regard d as a B module map d : P → A by composingwith the inclusion Ker b → B and the map B → A. Show that the ideal

I = a ∈ A : ad = 0

is independent of the choice of such (P, d) and that, regarding I as a Bmodule by restriction of scalars along B → A, we have a natural isomorphismI = Ext1B(H

0(L), A).(2) Show that Ext1B(L,A) ∼= A (this is Lemma 7.3.2 in the text), and that the

natural monomorphism

Ext1B(H0(L), A) → Ext1B(L,A)

is identied naturally with the inclusion I ⊆ A.

Hint: Let P• → H−1(L) be a projective resolution. It extends to give a projectiveresolution

· · · // P2d2 // P1

d1 // P0d0 // B

b // B // H0(L) // 0

of H0(L). Observe that

...d3 −10 d2

...d2 −10 d1

P2 ⊕ P1

0 1

0 0

//d2 1

0 d1

P1 ⊕ P0d1 1

0 d0

P1 ⊕ P0

0 1

0 0

//d1 −1

0 d0

P 0 ⊕Bd0 −10 b

P0 ⊕B

(d0 1

)

0 1

0 0

// B ⊕B

(b 1

)

Bb // B

is a projective Cartan-Eilenberg resolution of L whose horizontal cohomology givesthe above projective resolutions of the cohomology of L. Set P i := HomB(Pi, A).Forgetting the bottom row and applying HomB(_ , A), we obtain the following


double complex by transposing matrices and noting that b 7→ 0 under B → A.

......

P 1 ⊕ P 0

0 0

1 0

//

d∗2 0

−1 d∗1

OO

P 2 ⊕ P 1

d∗3 0

−1 d∗2

OO

P 0 ⊕ A

0 0

1 0

d0,1h

//

d∗1 0

1 d∗0

d0,1v

OO

P 1 ⊕ P 0

d∗2 0

1 d∗1

d1,1v

OO

A⊕ A

0 0

1 0

d0,0h

//

d∗0 0

−1 0

d0,0v

OO

P 0 ⊕ A

d∗1 0

−1 d∗0

d1,0v

OO

Set d0T = d0,0v +d0,0h and d1T = d1,0v −d0,1h to x sign conventions for the dierentials inthe total complex Tot (note the image of Tot in D(B) is RHom(L,A)). Now show:

Ker d0,1v = (f, a) ∈ P 0 ⊕ A : f + ad0 = 0= (ad0,−a) : a ∈ A∼= A

Ker d1T = (a1d0,−a1), (a2d0 − a1d0, a2) : a1, a2 ∈ A∼= A⊕ A

Im d0T = (ad0,−a), (0, a) : a ∈ A∼= A

Ext1B(L,A) = H1(Tot)

= (Ker d1T )/(Im d0T )∼= (A⊕ A)/(−a, a) : a ∈ A∼= A

We also have an exact sequence

0 // I // Ad∗0 // P 0

from the denition of I, and the inclusion

Ext1B(H0(L), A) → Ext1B(L,A)


corresponds to the inclusion I → H1(Tot) induced by including I = 0 ⊕ I in the(0, 1) term of the above double complex (note this is in Ker d0,1v ∩ Ker d0,1h and themap clearly induces a monomorphism on H1(Tot) because nothing of this form isin Im d0T ).

The E2 term of the spectral sequence where we rst take horizontal cohomologyis

Ep,q2 = Extp(H−q(L), A),

which clearly vanishes except when (p, q) ∈ [0,∞) × [0, 1]. The only interesting d2dierential is therefore:

dp,12 : Extp(H−1(L), A)→ Extp+2(H0(L), A).

(1) Show that this map is surjective when p = 0 and an isomorphism for p > 0.

Hint: Identify this map with the map on cohomology induced by the map ofcomplexes

A0 // A

d∗0 // P 0d∗1 // P 1

d∗2 // · · ·

P 0

Id

OO

d∗1 // P 1d∗2 //

Id

OO

· · ·obtained by applying Hom(_ , A) to the projection

· · · // P1d1 // P 0

· · · // P1d1 //

Id

OO

P 0

Id

OO

d0 // Bb // B.

The next several exercises concern Quot schemes and the deformation theory ofquotients (10).

Exercise 13. Let k be an algebraically closed eld, C a smooth proper curve overk, E a vector bundle on C. Let QuotE denote the scheme parameterizing (atfamilies of) short exact sequences

0→ S → E → Q→ 0

of sheaves on C and let Quotr,dE denote the component of QuotE where S hasrank r and degree d. Quotr,dE is projective. The subsheaf S is locally free sinceit is a subsheaf of a locally free (coherent) sheaf and OC is a sheaf of PIDs. Inthis exercise we will mostly be concerned with the spaces Qd,n := Quotn,−dOn

C andQn :=

⨿dQd,n.

(1) Show that the projective scheme Qd,n is smooth.


(2) On Qn, the determinant of the inclusion S → OnC gives an inclusion detS →

OC . The quotient of this inclusion determines a morphism

detn : Qn → QuotOC =⨿d

Symd C

restricting to a morphism detd,n : Qd,n → Symd C. Let P1, P2, . . . , Pl bedistinct points of C, and let m1, . . . ,ml be positive integers, so

∑imiPi is a

typical point of SymC. Show that

det−1(∑i

miPi) ∼=∏i

det−1(miPi)

(as schemes). Hint: Glue subsheaves over the open sets Ui := C \ (∪j =iPj).(3) Show that det−1n (dP ) does not depend on P or C, only on d and n, hence we

may denote it Xd,n. In fact, show that Xd,n is nothing but the quot schemeparameterizing quotients

⊕nk[t]/tM → Q

of k[t]/tM modules with dimk Q = d (this is clearly independent of thechoice of M ≥ d) discussed in (14). Hint: Let NM(P ) := SpecC OC/m

MP∼=

Spec k[t]/tM be the (M − 1)st innitesimal neighborhood of P in C. If

0→ S → OnC → Q→ 0

is a SES on C where S has rank n (so Q is torsion) and dimk H0(C,Q) =

d and Q is supported topologically on P , then on dimension grounds thequotient map factors (uniquely) through

OnC → On

NM (P )

as long as M ≥ d (Q is pushed forward from NM(P )).(4) Working over k = C, show that the Betti numbers of the smooth projective

variety Q2,2 (over the curve C = P1) are: b0 = 1, b2 = 2, b4 = 4, b6 = 2, b8 = 1.(The cohomology ring H∗(Q2,2,Q) is computed in Section 6 of: T. Braden,L. Chen, and F. Sottile, The equivariant Chow rings of Quot schemes.) Tryto understand the class of X2,2 in the Grothendieck ring of varieties in termsof that of Q2,2 by studying the stratication

Q2,2 = det−12,2(Sym2 P1 \∆)

⨿det−12,2(∆).

You should be able to at least compute the topological Euler characteristicof X2,2.

Exercise 14. Let k be an algebraically closed eld and let Xd,n denote the Quotscheme parameterizing (at families of) short exact sequences

0→ S → (k[x]/xM)n → Q→ 0

of k[x]/xM modules with dimk Q = d (for any suciently large M ; see below).

(1) Observe that, as suggested by the notation, this Quot scheme is independentof the choice of M as long as M ≥ d.


(2) Clearly Xd,n embeds in the Grassmannian Gr(dn − d, dn) of d dimensionalquotients of kdn by forgetting the k[x]/xd module structure. Express Xd,n →Gr(dn − d, dn) as a degeneracy locus of a map of vector bundles on theGrassmannian.

(3) Let λ = (λi) be a partition of d. Show that the locus Grλn in Xd,n wherethe quotient Q admits a direct sum decomposition Q ∼=

∑i Qi (as k[x]/xd

modules) with Qi indecomposable and dimk Qi = λi is locally closed in Xd,n.Show that the closure of Grλn in Xd,n is the union of the Grµn over partitionsµ ≤ λ (here ≤ is the usual" ordering of partitions, or perhaps the usual"ordering of the transposed partitions; the partition 1+ 1+ · · ·+1 should bethe minimum element in the ≤ ordering). My understanding is that Grλn iscalled the ane Grassmannian".

(4) Show that X1,n = Pn−1.(5) Show that Gr1+1+···+1

n∼= Spec k is a single point corresponding to the sub-

module S∗ of (k[x]/xd)n generated by

(x, 0, . . . , 0), (0, x, 0, . . . , 0), . . . , (0, . . . , 0, x),

whose quotient is Q∗ = kn.(6) Compute the dimension of the tangent space to Xd,n at the point

P = (0→ S∗ → (k[x]/xd)n → Q∗ → 0)

by using the isomorphism from deformation theory:

TPXd,n = Homk[x]/xd(S∗, Q∗).

(7) From (3) and (5), we have

X2,2 = Gr22⨿

Gr1+12 = Gr22

⨿Spec k.

Show that Gr22 is isomorphic to the total space of OP1(2). Hint: Show that,for any sequence

0→ S → (k[x]/x2)2 → Q→ 0

in Gr22, S can be generated as a k[x]/x2 module by a single element (this istrue of Q by denition of Gr22). If (a + bx, c + dx) is a generator of S, thenshow that the condition dimk S = 2 implies that at least one of a, c is ink∗. If, say, a ∈ k∗, then a + bx ∈ (k[x]/x2)∗, so S can be generated by anelement of the form (1, c + dx). Show that this denes an open embeddingA2

c,d → Gr22. Similarly, the locus where S can be generated by an element ofthe form (a+ bx, 1) denes an open embedding A2

a,b → Gr22. Show that theseoverlap in a Gm×A1, write down the transition function for the gluing, andrecognize it as the transition function for

OP1(2) = A2⨿

Gm×A1

A2.


(8) Conclude that X2,2 is a projective surface whose singular locus consists of asingle point P = Gr1+1

2 with

dimk TPX2,2 = dimk Homk[x]/x2(k2, k2) = 4.

(9) Consider the embedding X2,2 → Gr(2, 4). View X2,2 as Quot2(k[x]/x2)2 anduse (1, 0), (x, 0), (0, 1), (0, x) as an ordered k basis for k[x]/x2 and hence toidentify (k[x]/x2)2 with k4 as a k vector space. For 1 ≤ i < j ≤ 4, letUij∼= A4 ⊂ Gr(2, 4) be the usual chart for the Grassmannian centered at

⟨ei, ej⟩. The interesting chart (for our purposes) is U2,4∼= A4

t1,t2,t3,t4with

(t1, t2, t3, t4) corresponding to

⟨(t1, 1, t2, 0), (t3, 0, t4, 1)⟩ ∈ Gr(2, 4).

Give generators for the ideal I ∈ k[t1, t2, t3, t4] corresponding to the embed-ding

X2,2 ∩ U2,4 → U2,4∼= A4.

Notice that the singular point P ∈ X2,2 corresponds to the origin in A4, soshow that your ideal I denes a surface in A4 singular only at the origin witha 4 dimensional tangent there.

Exercise 15. Continue with the notation from (13), though we will always takeC = P1 here. The purpose of this exercise is to study a simple example whereQuotE is singular for a vector bundle E on P1.

(1) Working over C = P1, show that QuotE is smooth whenever E has rank atmost two (make use of Corollary 10.5.3 and the remark after it).

(2) In light of the aforementioned smoothness results, our search for a simple Ewith QuotE singular leads us to consider the bundle E := O2

P1 ⊕ OP1(−2).We will study X := Quot2,−2E for the remainder of this exercise. Writedown a point

P = (0→ S → E → Q→ 0) ∈ X(Spec k)

such thatdimk TPX = h0(P1, S∨ ⊗Q) = 5

and h1(P1, S∨ ⊗Q) = 1.(3) Recall the Quot scheme Q2,2 = Quot2,−2O2

P1 studied in the previous exercise.If

0→ S → O2P1 → Q→ 0

is a point of Q2,2 then show that the splitting type of S is either O(−1) ⊕O(−1) or O ⊕O(−2). The locus W ⊂ Q2,2 where S has the latter splittingtype is therefore closed by semicontinuity. Show that there is an isomorphismW = Sym2 P1 × P1 making the diagram

W

π1 ##GGG

GGGG

GG // Q2,2

det2

Sym2 P1


commute.(4) Similarly, if

0→ S → O2P1 → Q→ 0

is a point ofX then show that the splitting type of S is either O(−1)⊕O(−1)or O ⊕ O(−2), so that the locus Z ⊂ X where S has the latter splittingtype is closed by semicontinuity. Show that Z ∼= P1 × P3. Hint: P1 =PHom(O,O ⊕ O) and P3 = PHom(O(−2),O ⊕ O(−2)). On P1 × P3 × P1

we have the universal map from O to O ⊕O"i : O(−1, 0, 0) → O(0, 0, 0)2

f ⊗ s⊗ t 7→ sf(t)

pulled back via

π13 : P1 × P3 × P1 → PHom(O,O ⊕O)× P1.

Let q : O(0, 0, 0)2 → Q′ be the quotient of i. Q′ is an invertible sheaf(noncanonically) isomorphic to O(1, 0, 0). Similarly, we have a universalmap

O(0,−1,−2)→ O(0, 0, 0)⊕O(0, 0,−2)(pulled back via π23) with quotient r : O(0, 0, 0) ⊕ O(0, 0,−2) → Q. Showthat the composition

(r ⊗ IdQ′)

(q 0

0 IdQ′⊗π∗3O(−2)

): O(0, 0, 0)2 → Q⊗Q′

determines a morphism P1×P3 → Z and that this is in fact an isomorphism.(5) Show that X = Q2,2

⨿W Z.

Exercise 16. Recall that Quot1,d E ∼= PN is smooth for any vector bundle E onP1. Let C be a genus 2 curve and let E = OC ⊕ OC . We will see in this exercisethat Q := Quot1,−2 E is singular. To x notation, let L be the g12 on C (the onlyspecial degree two line bundle) and let p : C → P1 be the map obtained from L, soL = p∗OP1(1).

(1) Show that the map Q → Jac2C obtained from the dual of the universalsubbundle has ber P1 except over L where the ber is P3.

(2) Show that the singular locus of Q is a quadric in this P3 ber.

Exercise 17. Let A be a ring. Dene A algebras:

B1 = A[ϵ]/ϵ2

B′1 = A[ϵ]/ϵ3

B2 = A[x]/x2

B0 = B1 ⊗A B2

= A[ϵ, x]/(ϵ2, x2)

B = A[ϵ]/(ϵ3, x2)

F0 = A[y]/y2.


Let B′1 be the square zero A algebra extension

B′1 = 0 // Aϵ2 // B′1 / / B1

// 0

A

OO``BBBBBBBB

and let

B = 0 // A[x]/x2 ϵ2 // Bg // B0

// 0

A

OO__@@@@@@@@

be the algebra extension obtained from B′1 by applying ⊗AB2 (note that everythingin sight is at as an A module).

View F0 as a B0 module by restriction of scalars along the surjective A algebramap f : B0 → F0 sending both ϵ and x to y. Let N0 ⊆ B0 be the kernel of f , so wehave an exact sequence

0 // N0// B0

f // F0// 0.

Note that F0 is at over B1 (in fact it is free of rank one and naturally identiedwith B1 as a B1 module via the composition of B1 → B0 and f). Explicitly describethe obstruction

ω ∈ Ext1B0(N0, A[x]/x

2 ⊗B0 F0)

to lifting f to a surjection f : B → F with F at over B′1. Note that

A[x]/x2 ⊗B0 F0 = A⊗B1 F0

= A⊗B1 B1

= A

so we should regard ω as an element of Ext1B0(N0, A). In particular, show that ω = 0.

Hint: Let L ⊂ B be the kernel of the composition

Bg // B0

f // F0

so that the map g|L : L→ B0 clearly lands in (factors through) N0 ⊂ B0. Obviouslythe kernel A[x]/x2 of B → B0 is contained in L (via the map 1 7→ ϵ2 as in thediagram dening B), and in fact we have an exact sequence as in the top row of thediagram of B modules below:

0 // A[x]/x2 ϵ2 //

x7→0

Lg|L //

N0// 0

0 // A // M // N0// 0


The bottom row is dened by pushing out the top rowit denes the obstructionω (note that it is straightforward to see that M is a B0 module (i.e. ϵ2M = 0), sothis B module extension is actually a B0 module extension). You should nd thatM → N0 admits both a B1 linear section and a B2 linear section, but no B0 linearsection.

Exercise 18. Let X be a scheme, p : L → X a line bundle, n ∈ N, Y the nth

innitesimal neighborhood of the zero section in L. Note that π := p|Y : Y →X is an l.c.i. morphism. Describe the cotangent complex of π using the givenfactorization of π as a regular closed embedding Y → L followed by the smoothprojection p. (This is a twisted version of the cotangent complex of A → A[x]/xn

studied in 7.3.) If X → B is another map of schemes, show that the cotangentcomplex of Y → B splits as the sum of the cotangent complex of X → B and thecotangent complex of Y → X. Use this to descibe the cotangent complex of Y (overk) when X is a smooth scheme over B = Spec k, k a eld (say). Specialize to thecase X = P1

k, L = O(a). Specialize to the case a = −1, n = 1. Conclude thatExt1(LY ,OY ) = 0 in this case.

References

[B] N. Bourbaki, Algèbre. Masson, Paris, 1980.[BB] M. Barakat and B. Bremer, Higher extension modules and the Yoneda product.

arXiv:0802.3179.[CdJ] B. Conrad and J. de Jong, Approximation of versal deformations.[DM] P. Deligne and D. Mumford, The irreducibility of the moduli space of curves.[EE] E. Enochs and S. Estrada, Relative homological algebra in the category of quasi-coherent

sheaves. Adv. Math. 194 (2005) 284-295.[Eis] D. Eisenbud, Commutative Algebra with a view towards algebraic geometry.[EGA] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique. Pub. Math.

I.H.E.S., 1960.[Fog] J. Fogarty, Algebraic families on an algebraic surface. Amer. J. Math 90 (1968) 511-521.[Ful] W. Fulton, Intersection theory.[G] W. D. Gillam, Localization of ringed spaces. Adv. Pure Math. 1(5) (2011) 250-263.[Gro] A. Grothendieck, Techniques de construction et théorèms d'existence en géométrie al-

gébriques IV: les schémas de Hilbert. Sém. Bourbaki, 1960-1961 221 249-276.[H] R. Hartshorne, Algebraic Geometry. Springer, 1977.[H2] R. Hartshorne, Deformation theory Springer GTM 257, 2010.[IK] I. Kaplansky, Projective modules. Ann. Math. 68 (1958) 372-377.[Ill] L. Illusie, Complexe cotangent et déformations, I. Lec. Notes Math. 239. Springer-Verlag,

1971.[KS] K. Kodaira and D. Spencer, On deformations of complex-analytic structures, I, II. Ann.

Math. 67 (1958) 328-466.[L] D. Lazard, Autour de la platitude. Bull. Soc. Math. France, 97 (1969) 81-128.[LM] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik 39,

Springer-Verlag, Berlin, 2000.[Mat] H. Matsumura, Commutative Algebra. Second Ed. Benjamin/Cummings Pub. Co. 1980.[Mat2] H. Matsumura, Commutative ring theory.[Milne] J. Milne, Étale Cohomology. Princeton Univ. Press, 1980.[OS] M. Olsson and J. Starr, Quot functors for Deligne-Mumford stacks[O] F. Oort, Yoneda extensions in abelian categories. Math. Ann. 153(3) (1964) 227-235.


[RD] R. Hartshorne, Residues and duality.[SGA1] A. Grothendieck et al. Séminaire de Géométrie Algébrique du Bois Marie - 1960-

61: Revêtements étales et groupe fondamental (SGA 1) Lec. Notes Math. 224 (1971),Springer-Verlag, New York, Berlin.

[SGA3] M. Demazure and A. Grothendieck, eds. Séminaire de Géométrie Algébrique du BoisMarie - 1962-64: Schémas en groupes (SGA 3) Lec. Notes Math. 153 (1970), Springer-Verlag, New York, Berlin.

[SGA4] M. Artin, A. Grothendieck, and J. L. Verdier, eds. Séminaire de Géométrie Algébriquedu Bois Marie - 1963-64: Théorie des topos et cohomologie étale des schémas (SGA 4)(Volumes I,II,III) Lec. Notes Math. 269, 270, 305 (1972), Springer-Verlag, New York,Berlin.

[SGA6] P. Berthelot, A. Grothendieck, and L. Illusie, eds. Séminaire de Géométrie Algébrique duBois Marie - 1966-67: Théorie des intersections et théorème de Riemann-Roch (SGA 6)Lec. Notes Math. 225 (1971), Springer-Verlag, New York, Berlin.

[SGA7] P. Deligne, A. Grothendieck, N. Katz, et al. Séminaire de Géométrie Algébrique du BoisMarie - 1967-69: Groupes de monodromie en géométrie algébrique (SGA 7) (VolumesI,II) Lec. Notes Math. 288, 340 (1972,1973) (particularly Rim's deformation theoryarticle written in English)

[SP] A. J. de Jong, et al, The Stacks Project.[T] A. Grothendieck. Sur quelques points d'algèbra homologique. Tôhoku Math. J. 9 (1957)

119-221.[Vis] A. Vistoli, Notes on Grothendieck topologies, bered categories, and descent theory.[Y] N. Yoneda, On Ext and exact sequences. J. Fac. Sci. Univ. Tokyo 8 (1960) 507-576.

ETH Zürich, Departement Mathematik, Rämistrasse 101, 8092 Zürich

E-mail address: [email protected]

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