FREIE UNIVERSITÄT BERLINÉCOLE NORMALE SUPÉRIEURE DE RENNES

Low-degree covers in algebraic geometry

Gabriel LEPETIT

Thesis of first year of Master, written under the supervision of Fabio TONINI

MAGISTÈRE 2 2015-2016

This document has been redacted from May 2nd to July 1st 2016 in the :

Freie Universität Berlin - Mathematisches InstitutArnimallee 314195 Berlin

My advisor was Fabio TONINI (http://www.mi.fu-berlin.de/users/tonini/), re-searcher in the Arithmetische Geometrie team directed by Hélène ESNAULT (http://www.mi.fu-berlin.de/users/esnault/).

1

Contents

Introduction 3

1 Sheaves and schemes 41.1 Sheaves of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Quasi-coherent sheaves of modules . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Locally free sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Relative spectrum of a quasi-coherent sheaf of algebras . . . . . . . . . . . . . . 131.6 Symmetric algebra and exterior powers of algebra . . . . . . . . . . . . . . . . . 18

2 Double covers 202.1 Basic facts about covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Double covers via line bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Triple covers 283.1 The local case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 The global case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Equivalence between Cov3(Y ) and eD3(Y ) . . . . . . . . . . . . . . . . . . 313.2.2 Equivalence between eD3(Y ) and D3(Y ) . . . . . . . . . . . . . . . . . . . 34

Bibliography 38

◊ ◊ ◊ ◊

Acknowledgements

First of all, I would like to address special thanks to my advisor, Fabio Tonini, who patientlyanswered my questions and enthusiastically introduced me to the world of algebraic geom-etry. I thank also Hélène Esnault who accepted to receive me in Berlin, and all the membersof the team and guests with whom I had the chance to have fruitful discussions, in particularLei Zhang, Marta Pieropan and Marco d’Addezio. I am also grateful to Lars Kindler and hiswonderful coffee machine which deliciously boosted my productivity.

2

INTRODUCTION

This thesis aims to define a simple notion of cover in algebraic geometry that would rea-sonably look like the one we know in topology and that is central in differential geometryand algebraic topology. In this context, if Y is a topological space, a cover of Y is a continu-ous map f : X → Y such that for all y ∈ Y , there exists an open neighbourhood V of y anda non-empty set I y such that f −1(V ) =

⊔

i∈I y

Ui and f|Uiis a homeomorphism from Ui onto V .

We know that if Y is connected, and if the fibers are finite sets, then they all have the samenumber of elements which is the common cardinality of the sets I y , y ∈ Y . This is what wecall an unramified cover : the copies of Y produced by the cover don’t cross. We would liketo have a similar algebraic notion, that would allow ramification, as in the example 2.1.5 :the cardinality of the fibers would still be the same, but only if we count the elements withmultiplicity.

The notion of cover plays an important role in the current research in algebraic geometry,for several reasons. First, it is a convenient way to build new spaces – for example algebraicvarieties – out of the ones we already know. Moreover, in the theory of moduli spaces, thestudy of the Hurwitz spaces (see [12]), which are parametrizing covers of curves, have beenan active field of research for the last years. The Hurwitz spaces are not simply studied assets of covers, but as schemes whose closed points are the covers of curves. The coversare often studied together with group actions, especially in the theory of Galois covers (cf[11]). For instance, Z/2Z acts on the double covers, but this situation does not recur forhigher-degree covers, as Z/3Z does not act on every 3-cover.

A cover of a scheme Y is simply defined as an affine morphism of schemes f : X → Ysuch that f∗OX is a locally free OY -algebra of finite rank (cf 2.1.1). We will first presentbriefly in the first chapter the prerequisites of algebraic geometry necessary to understandthe study of covers, in particular the quasi-coherent sheaves of algebras.

In the second and third chapters, we give a full description of the categories of coversof degree 2 and 3. The goal is to extract the essential data of the multiplication map, andthe trace map for sheaves of algebras (proposition 2.1.6) play a central role in this study.It happens that the covers of degree 1 are the isomorphisms (example 2.1.4), the ones ofdegree 2 correspond classically to the pairs (L ,σ) where L is a locally free sheaf of rank 1and σ is a morphism from L ⊗L to OY (theorem 2.2.1), and the 3-covers, as described in[9], are similar to the (E ,δ)where E is a locally free OY -module of rank 2 and δ : S3E → Λ2Eis a morphism (theorem 3.0.7).

For higher degree, it is not possible anymore to have a general description of the coverswith locally free sheaves and maps without further relations, but G. CASNATI and T. EKDAHL

([3] and [2]) developed the theory of Gorenstein covers, leading to a particular study of the4 and 5-covers, while in [7], an analysis of the quadruple covers of algebraic varieties wasdone. Rita PARDINI ([10]) studied the triple covers in characteristic 3, a case where ourmethod doesn’t work. The work of M. BOLOGNESI and A. VISTOLI ([1]) is another relevantarticle involving the triple covers.

3

Chapter 1

Sheaves and schemes

In all the thesis, I will assume that the basic facts of commutative algebra are known, anduse freely the vocabulary of the theory of categories. The main results we use are summedup in [6] pp 541-545 (categories), [8], chapter 1 (tensor product and localization).

The main reference for this chapter is the book of Qing Liu ([8]), chapter 2.

1.1 Sheaves of rings

Definition 1.1.1

A presheaf of sets on the topological space X is a contravariant functor F from thecategory of the open subsets of X into the category of sets. More concretely, it is the dataof a set F (U), for every U open subset of X , and, for V ⊂ U open subsets, restrictionmaps ρUV :F (U)→F (V ) satisfying the following axioms :

• For all W ⊂ V ⊂ U , ρUW = ρVW ρUV .

• ρUU = idU .

The elements of F (X ) are called global sections.Example 1.1.2. The most intuitive example of a presheaf is the presheaf of rings on XF : U 7→ C 0(U ,R), endowed with the restrictions ρUV : f 7→ f|V , where C 0(U ,R) is the setof the continuous fonctions from U to R.

Because of this example, we denote, for an arbitrary presheaf, ρUV (s) = s|V .

Definition 1.1.3

A sheaf on X is a presheaf on X F satisfying the following axioms :

• (Uniqueness)For U open of X and U =⋃

i∈IUi open covering of U , if s, t ∈ F (U) are

such that ∀i ∈ I , s|Ui= t|Ui

, then s = t.

• (Glueing) For U open of X and U = ∪i∈I Ui open covering of U , if for every i ∈ I ,si ∈ F (Ui) and ∀i, j ∈ I , si|Ui∩U j

= s j|Ui∩U j, then there exists a s ∈ F (U) such that

∀i ∈ I , s|Ui= si.

Definition 1.1.4

A morphism of presheaves of sets α : F → G is a family of morphisms F (U)α(U)−→ G (U)

compatible with the restrictions : for every V ⊂ U ⊂ X , the following diagram commutes.

4

F (U) G (U)

F (V ) G (V )

α(U)

ρFUV ρGUVα(V )

If F and G are sheaves, we say that α is a morphism of sheaves.

Remark. We can also consider presheaves of groups, abelian groups, rings, A-modules orA-algebras where A is a fixed ring. In each case, the restriction maps are morphisms in theconsidered category, as well as the morphisms of presheaves.

We see that a sheaf if a way to pass from the local data to the global one.Definition 1.1.5

Let F be a sheaf on X and x ∈ X . The stalk of F at x is

Fx = lim−→x∈U

F (U)

where the limit is taken on the inductive system (for the inclusion) of the open subsetsof X containing x .

Concretely, in Fx , any element is defined on a open neighbourhood of x and two ele-ments are identified if they coincide on an open neighbourhood of x .

We can also take the stalk at x of a morphism α :F →G , by setting αx : sx → (α(U)(s))xif s ∈ F (U). αx is a morphism.Proposition 1.1.6

Let F be a sheaf on X , U ⊂ X an open subset, and s, t ∈ F (U). Then s = t ⇔ ∀x ∈U , sx = t x .

Proposition 1.1.7

If α :F →G is a morphism of sheaves on X , then α is an isomorphism if and only if αx

is an isomorphism for all x ∈ X .

The following proposition allows us to define a sheaf starting with a presheaf, and thisoperation, called sheafification, preserve the stalks and the morphisms.Proposition 1.1.8 (sheafification)

Let F be a presheaf on X . There exists a unique sheaf F † on X , together with a mor-phism θ :F →F † satisfying the following universal property :

For every u :F →G morphism, where G is a sheaf, there exists a unique morphismeu :F †→G such that u= eu θ .

Moreover, for all x ∈ X , θx is an isomorphism between Fx and F †x .

If Psh(X ) (resp. Sh(X )) is the category of the presheaves (resp. sheaves) on X , sheafifi-cation defines a functor ·† : Psh(X ) −→ Sh(X )

F 7−→ F †. If u is a morphism of presheaves from

F to G , u† is the unique morphisme making this diagram commute :

F G

F † G †

u

θF θG

u†

5

Let us remark that the family of morphisms (θF ) define a natural transformation betweenidPsh(X ) and i ·† where i : Sh(X )→ Psh(X ) is the inclusion functor.

We can keep in mind for the rest of the thesis that, since the morphisms θ preserve thestalks, it is not a problem to define a morphism between presheaves if we want to study thesheafified morphism.

We will now explain how it is possible to define a sheaf on a basis of open subsets (see[5], pp. 25-28, for more details).

Let X be a topological space, and B a basis of open subsets of X (i.e. for all open U ofX , there exist elements Ui ofB such that U =

⋃

i Ui).AB-presheaf of sets (resp. groups, rings) on X is a contravariant functor fromB , seen

as a subcategory of the category of the open subsets of X , to the category of sets (resp.groups, rings).

Definition 1.1.9

AB-sheaf on X is aB-presheaf F satisfying the two following axioms :

• (Uniqueness) If U is an element of B such that U =⋃

i∈IUi, Ui ∈ B , and s ∈ F (U)

is such that s|Ui= 0∀i ∈ I , then s = 0.

• (Glueing) If U is an element of B such that U =⋃

i∈IUi, Ui ∈ B , if for every i ∈ I ,

si ∈ F (Ui) and ∀i, j ∈ I ,∀W ∈B such that W ⊂ Ui ∩U j, we have si|W = s j|W , thenthere exists a s ∈ F (U) such that ∀i ∈ I , s|Ui

= si.

We can define stalks of a B-presheaf the same way we did with presheaves, as well asmorphisms ofB-presheaves.

Proposition 1.1.10

Let F 0 be aB-sheaf on X . Then there exists a unique (up to isomorphism) sheaf F onX such that F (U) =F 0(U) for every U inB . The sheaf F is defined by

∀V ⊂ X ,F (V ) = lim←−U∈BU⊂V

F (U)

Moreover, if x ∈ X , then Fx =F 0x .

Remark. Thanks to the universal property of the projective limit, we can also extend amorphism ofB-sheaves in a unique way in a morphism between the corresponding sheaves.Moreover, the stalks of these two morphisms will coincide.

Therefore, this construction gives an equivalence between the categories of the B-sheaves on X and of the sheaves on X .

If f : X → Y is a continuous map, the inverse and direct image functors are used to passfrom a sheaf on X to a sheaf on Y .

• If (F ,ρ) is a sheaf on X , the direct image f∗F of F is the sheaf defined by f∗F (V ) =F ( f −1(V )) for all V ⊂ Y with the restrictions µV V ′ = ρ f −1(V ) f −1(V ′).

If α : F → G is a morphism of sheaves on X , then f∗α : f∗F → f∗G is the morphismof sheaves defined by ( f∗α)(V ) = α( f −1(V )) for all V ⊂ Y .

6

We have a morphism : ( f∗F ) f (x) −→ Fx

s f (x) 7−→ sx

but it isn’t injective in general. But for

instance if f induces a homeomorphism from X into its image, then this morphismwill be an isomorphism.

• If G is a sheaf on Y , the inverse image f −1G of G is the sheaf defined by :

∀U ⊂ X ,

f −1G

(U) = lim−→f (U)⊂V

G (U)

We define the restrictions using the action of the inductive limits on the morphisms,and likewise taking the inverse image of a morphism of sheaves makes sense.

What we should keep in mind is that if x ∈ X , then ( f −1G )x = G f (x), and if i : V ,→ Yis an inclusion, then i−1G is the restricted sheaf G|V : V ′ ⊂ V 7→ G (V ′).

Finally, we will quite often use exact sequences of sheaves. We should underline thefact that if γ is a morphism of sheaves of groups, U 7→ Im (γ(U)) is only a presheaf andnot a sheaf. That is why Imγ is defined as the sheafification of this presheaf. Besides,U 7→ ker(γ(U)) is a sheaf called kerγ.

Definition 1.1.11

A sequence of morphisms of sheaves Fα−→G

β−→H is said to be exact if kerβ = Imα.

Proposition 1.1.12

A sequence F → G → H of sheaves on X is exact if and only if for all x ∈ X , thesequence Fx →Gx →Hx is exact.

1.2 SchemesDefinition 1.2.1

A locally ringed space is a pair (X ,OX ), where X is a topological space and OX is a sheafof rings on X such that for all x ∈ X , OX ,x := (OX )x is a local ring. If its maximal ideal ismx , then the field k(x) := OX ,x/mx is called the residue field of X at x.

Definition 1.2.2

A morphism of locally ringed spaces f : X → Y is f = ( f , f #) where f is a continuousmap between X and Y , and f # : OY → f∗OX is a morphism of sheaves such that for everyx ∈ X , f #

x = OY, f (x) −→ ( f∗OX ) f (x)can−→ OX ,x is a local ring morphism.

We will omit the OX in the notation for simplicity.

Remark. • If f : X → Y and g : Y → Z are morphisms of locally ringed spaces, thenf g is ( f g, f∗g

# f #).

• If U is an open subset of X and i : (U ,OX |U)→ (X ,OX ) is the inclusion morphism, with

∀V ⊂ X , i#(V ) : OX (V ) −→ (i∗OX |U)(V ) = OX (U ∩ V )s 7−→ s|U∩V

then the restriction of a morphism f : (X ,OX )→ Y to U is f|U = f i.

7

Let A be a ring. We want to put a structure of locally ringed space on X = Spec(A), setof the prime ideals of A.

Let us recall briefly that we can define a topology on X , called the Zariski topology,whose closed subsets are the V (I) = p ∈ X : I ⊂ p, I ideal of A. The open subsets D( f ) =X \V ( f A), f ∈ A are called principal open subsets and form a basisB of open subsets of X .

We set OX (D( f )) = A f , localization of A at f . This definition makes sense, since ifD( f ) = D(g), A f ' Ag . If D( f g) ⊂ D( f ), the restriction map is given by A f → A f g = (A f )g .The association OX is aB-sheaf and therefore can be extended to a sheaf on X . We call OX

the structural sheaf of X .

Furthermore, if x = p ∈ X , then OX ,p = Ap, localization of A by A\ p.See [8], p.17, for a more detailed proof.

Definition 1.2.3

• An affine scheme is a locally ringed topological space that is isomorphic to someX = Spec A endowed with his structural sheaf.

• A scheme is a locally ringed space (X ,OX ) such that there exists a open coveringX =

⋃

i∈IUi of X such that for all i ∈ I , (Ui,OX |Ui

) is an affine scheme.

If X is a scheme, an open subset U such that the open subscheme (U ,OX |U) is affine issaid to be open affine.

Finally, we will study the morphisms of schemes, i.e. the morphisms of locally ringedspaces between schemes.

Definition 1.2.4

An open embedding is a morphism i : X → Y such that i is a topological open embedding(i.e. induces an homeomorphism from X onto i(X ) open), and ∀x ∈ X , i#

x : OY,i(x)→OX ,x

is an isomorphism.

Remark. A morphism i : X → Y is an open embedding if and only if there exists V opensubset of Y such that i induces an isomorphism of schemes between (X ,OX ) and (V,OY |V ).

Proposition 1.2.5

If u : A→ B is a morphism of rings, then there exists a morphism fu : Spec B → Spec Asuch that f #

u (Spec A) = u.

Theorem 1.2.6

Let Y be an affine scheme.Then for every scheme X , there exists a canonical bijection

ψX : Homsch.(X , Y ) −→ Homrings(OY (Y ),OX (X ))ϕ 7−→ ϕ(X )

This family of maps defines a natural transformation between the functors X 7→Hom(X , Y ) and X 7→ Hom(OY (Y ),OX (X )) : if α : Z → X is a morphism of schemes,then we have

8

Hom(X , Y ) Hom(OY (Y ),OX (X ))

Hom(Z , Y ) Hom(OY (Y ),OZ(Z))

ψX

ψZ

α#(Z)··α

The key argument of the proof is to use the previous proposition when X is an affinescheme, and to reduce to this case by taking an affine covering when X is an arbitraryscheme.

Remark. In particular there is an anti-equivalence of categories between the category of theaffine schemes and the category of the rings.Definition 1.2.7

If X and Y are schemes, Y is said to be a X -scheme if it is given a morphism α : Y → X ,called structural morphism. If X = Spec A is affine, we also say that Y is a A-scheme.

A morphism f : Y → Y ′ of X -schemes is a morphism of schemes such that :

Y Y ′

X

f

α′α

commutes. The set of the morphism of X -schemes between Y and Y ′ is denoted byHomX (Y, Y ′).

It is easy to see with the theorem 1.2.6, that if A is a ring, then the category of affineschemes over A is equivalent to the category of A-algebras.

1.3 Quasi-coherent sheaves of modules

For this section, we refer to [8], chapter 5, pp. 158-163.Definition 1.3.1

Let X be a scheme. A OX -moduleF is a sheaf on the topological space X such that for allU ⊂ X open, F (U) is a OX (U)-module and if V ⊂ U , the restriction ρUV is a morphismof OX (U)-modules for the structure of OX (U)-module on F (V ) defined by a.s = a|V .s.

Remark. A morphism of OX -modules is simply a morphism of sheaves ϕ such that for everyU ⊂ X , ϕ(U) is a morphism of OX (U)-modules.

As for the modules over a ring, we can define the tensor product and the direct sum ofsheaves of OX -modules :

• If F and G are two OX -modules, we define a presheaf H : U 7→ F (U)⊗OX (U) G (U)endowed, for V ⊂ U with the restrictions ρFUV ⊗ ρ

GUV : s ⊗ t 7→ s|V ⊗ t|V . Clearly, if

x ∈ X , the stalk ofH at x isHx =Fx ⊗OX ,xGx .

The tensor product F ⊗OXG of F and G is the sheafification ofH . According to the

proposition 1.1.8, we have

(F ⊗OXG )x =Hx =Fx ⊗OX ,x

Gx

Using the universal property of the sheafification, we will often rather give the de-scription of morphisms onH than define them on F ⊗OX

G .

9

• If (Fi)i∈I is a family of OX -modules, then the OX -module⊕

i∈IFi is the sheafification of

the presheaf defined by

∀U ⊂ X ,

⊕

i∈I

Fi

(U) =⊕

i∈I

Fi(U)

the direct sum on the right being a direct sum of OX (U) modules.

Definition 1.3.2

Let X be a scheme and F be a OX -module. We say that F is quasi-coherent if for allx ∈ X , there exists an open neighbourhood U of x and an exact sequence of OX -modules

O (J)X |U →O(I)X |U →F[U → 0

This concept is very convenient, as there is an equivalence between the category ofquasi-coherent OSpec A-modules and the category of A-modules, given by the operation ofsheafification of a module.Proposition 1.3.3

Let A be a ring, and M be a A-module. Then, if X = Spec A, we can define a OX -moduleeM by setting, for every principal open subset D( f ) ⊂ X , eM(D( f )) = M f

Moreover, if x = p ∈ Spec A, then eMx = Mp.

Remark. • The operation of sheafification is similar to the construction of the affineschemes p. 8.

• This transformation is compatible with the direct sum and the tensor product :

ã

⊕

i∈I

Mi =⊕

i∈I

fMi and åM ⊗A N = eM ⊗OXeN

Proposition 1.3.4

With the same notations, a sequence of A-modules L → M → N is exact if and only ifthe sequence of OX -modules eL → eM → eN is exact. Thus if M is a A-module, then eM isquasi-coherent.

Theorem 1.3.5

If X is a scheme and F is a OX -module, then F is quasi-coherent if and only if for all Uopen affine, F|U =áF (U).

Corollary 1.3.6

Let QCoh(Y ) be the category of quasi-coherent sheaves on the scheme Y . If Y = Spec Ais affine, then there is a equivalence of categories between QCoh(Y ) and ModA given bythe functors ∆ :F 7→F (Y ) and Λ : M 7→ eM .

10

1.4 Locally free sheaves

Let X be a scheme.

Definition 1.4.1

• A free sheaf of rank n on X is a sheaf F on X that is isomorphic to O nX .

• A sheaf F is said to be a locally free sheaf of rank n if there exists a coveringX =

⋃

i∈IX i of X by open subsets such that for every i ∈ I , F|X i

is free of rank n.

• A sheaf F is locally free of finite rank if there exists a covering X =⋃

i∈IX i together

with integers ni such that F|X iis locally free of rank ni.

A locally free sheaf of rank 1 is called an invertible sheaf, or a line bundle.

Remark. If F is a free sheaf of rank n on X , then if (α1, . . . ,αn) is a basis of the OX (X )-module F (X ), for every U ⊂ X , (α1|U , . . . ,αn|U) is a basis of F (U). We often say that (αi) isa basis of F .

An immediate consequence of this fact is that we can check the equality of two mor-phisms F →G , F free of finite rank, on F (X ).

Moreover, a morphism α between two free sheaves on X is entirely determined by thematrix of α(X ), also denoted by M(α).

Definition 1.4.2

Let F be a quasi-coherent OX -module. It is said to be finitely presented if for every Uopen affine subset of X , F (U) is a finitely presented OX (U)-module.

Proposition 1.4.3

Let F be a finitely presented quasi-coherent sheaf on X . Then, if n ∈ N∗, F is locallyfree of rank n if and only if ∀x ∈ X , Fx is a free OX ,x -module of rank n.

Lemma 1.4.4

Let A be a ring, M , N two finitely presented A-modules and p ∈ Spec A. If u : M → N is

such that Mp

up

' Np for p ∈ Spec A, then there exists f ∈ A\ p such that M f

u f' Nf .

Proof. First, let us notice that coker u= N/Im u is finitely generated since N is of finite type.The exact sequence 0 → ker u → M

u→ N → coker u → 0 yields, by flatness of the

localization, 0→ ker up→ Mp

u→ Np→ coker up→ 0.

But, by hypothesis, coker up = 0, so ∀x ∈ coker u,∃ fx ∈ A\p : fx x = 0. Taking generatorsx1, . . . , xn of coker u and setting f = fx1

. . . fxn, we obtain ∀x ∈ coker u, f x = 0, f ∈ A \ p.

Finally, coker u f = (coker u) f = 0 implies that u f is surjective.Thanks to [13, Tag 0517], also ker u is finitely generated and likewise, there exists f ′ ∈

A\ p such that ker u f ′ = 0. Therefore, u f f ′ : M f f ′ → Nf f ′ is an isomorphism.

Proof (of 1.4.3). Let us assume that Fx is free of rank n for all x ∈ X . We can assume thatX is affine since we want to prove a local statement.

We have Fψ' eM , where M is a finitely presented OX (X )-module. Therefore we have an

isomorphism Fx ' eMx .

11

Composing with an isomorphism betweenFx and O nX ,x , we obtain ϕx : O n

X ,x → eMx . SinceO n

X is free, and replacing X by an open affine V if necessary, we can find γ morphism fromO n

X to eM such that γx = ϕx .

Thus, γx : O nX ,x ' Mp and γx is the localization at x of γ(X ) as X is affine. According

to the lemma, there exists f ∈ OX (X ) such that γ(X ) f : OX (X )nf → M f is an isomorphism.

Therefore, as D( f )' SpecOX (X ) f is an open affine, eM|D( f )→O nX |D( f ).

Remark. If F is a locally free sheaf of finite rank, then we call rank of F at x ∈ X theinteger rankxF = rankOX ,x

Fx . The function x 7−→ rankxF is locally constant. Therefore, ifX is connected, it is constant of value d, so F is locally free of rank d.

Definition 1.4.5

Let F ,G be two sheaves (of abelian groups, rings, of OX -modules...) on X . The sheafHom(F ,G ) is defined by

∀U ⊂ X ,Hom(F ,G )(U) = Hom(F|U ,G|U)

In particular, if F is a OX -module, the dual sheaf of F is F ∨ := Hom(F ,OX ).

Proposition 1.4.6

Let E ,E ′ be locally free sheaves of finite rank. Then Hom(E ,E ′) is a locally free sheaf offinite rank.

Theorem 1.4.7

If Y is a scheme and H is a locally free sheaf on Y of finite rank, F and G are quasi-coherent sheaves on Y , then we have a canonical bijection

Hom(F ⊗H ,G )uH−→ Hom(F ,G ⊗H ∨)

Proof. First of all, the canonical morphism p : H ⊗H ∨ −→ End(H )s⊗ϕ 7−→ t 7→ ϕ(t)s

turns out to

be an isomorphism, since H is locally free of finite rank and the corresponding map fora free module on a ring is classically an isomorphism. So, composing with the structuralmorphism ι : OY −→ End(H )

1 7−→ idH, we define :

OY H ⊗H ∨

End(H )

u

p

Ýι

We also have the evaluation morphism ev : H ⊗H ∨ −→ OY

s⊗ϕ 7−→ ϕ(s).

If α :F ⊗H →G , we set uH (α) :Fu−→F ⊗H ⊗H ∨ α⊗id

−→G ⊗H ∨.

Given a morphism β :F →G ⊗H ∨, we set vH (β) =F ⊗Hβ⊗id−→ G ⊗H ∨ ⊗H

id⊗ev−→ G

and we want to prove that vH is the inverse of uH .

Given α :F ⊗H →G and β = uH (α), then, if U is an open subset of Y and u(U)(1) =∑

j h j ⊗ϕ j, h j ∈H ,ϕ j ∈H ∨, γ= vH (β) is, on U :

12

F ⊗H F ⊗H ⊗H ∨ ⊗H G ⊗H ∨ ⊗H G

f ⊗ h∑

j f ⊗ h j ⊗ϕ j ⊗ h∑

j α( f ⊗ h j)⊗ϕ j ⊗ h∑

jϕ j(h)α( f ⊗ h j)

id⊗u⊗id α⊗id⊗id id⊗ev

We want to show that for every f ∈ F (U), h ∈ H (U),∑

jϕ j(h)α( f ⊗ h j) = α( f ⊗ h).This statement can be checked on the basis of open subsets B constituted by the U openaffine such that H|U is free. To simplify, let us assume that U = Y , and let us fix a basis(v j) j=1...r ofH (Y ). Then the dual family (v∗j ) j=1...r is a basis ofH ∨(Y ).

As p(Y )

∑

j v j ⊗ v∗j

=

h 7→∑

j v∗j (h)v j = h

= idH (Y ), we have u(Y )(1) =r∑

j=1v j ⊗ v∗j .

Consequently,

∀ f ∈ F (Y ),∀i ∈ J1; rK,γ(Y )(vi) =∑

j

α(Y )( f ⊗ v j)v∗j (vi) = α(Y )( f ⊗ vi)

and by linearity, we have the desired equality and vH uH = id.

We show that uH vH = id using a similar method.

1.5 Relative spectrum of a quasi-coherent sheaf of alge-bras

Definition 1.5.1

Let Y be a scheme. A sheaf of algebras A on Y , or OY -algebra, is a OY -module, whereA (U) is endowed with a structure of ring that is compatible with the structure of OY (U)-module.

There are several useful equivalent ways to see a OY -algebra :

• It is a sheaf of rings A together with a morphism of sheaves, called structural mor-phism, α : OY → A . The structure of OY (U)-module on A (U) is given by ∀a ∈OY (U),∀ f ∈A (U), a. f = α(a) f .

• It is a OY -module A , endowed with a multiplication morphism of OY -modules m :A ⊗A → A which has a neutral element 1A and satisfies the commutativity andassociativity diagrams (the swap map is just s⊗ t 7→ t ⊗ s) :

A ⊗A A ⊗A

A

m m

swap

and

(A ⊗A )⊗A A ⊗ (A ⊗A )

A ⊗A A ⊗A

A

id

m⊗id

m

id⊗m

m

Remark. Seeing the commutativity and associativity of m as the commutativity such dia-grams makes clear the fact that these properties are equivalent to the corresponding localdiagrams (i.e. on the stalks).

13

Definition 1.5.2

A morphism of OY -algebras φ :A →B is, in a equivalent way :

• A morphism φ of OY -modules such that φ(1A ) = 1B and

A ⊗A B ⊗B

A B

φ⊗φ

m nφ

• A morphism of sheaves of rings such that

A B

OY

φ

α β

A quasi-coherent sheaf of algebras is simply a sheaf of algebras that is quasi-coherent asa sheaf of modules. The category of quasi-coherent OY -algebras is denoted by QAlg(Y ).

Definition 1.5.3

Let X and Y be two schemes. A morphism f : X → Y is called affine if for every V openaffine of Y , f −1(V ) is affine.

We call Aff(Y ) the category of affine morphisms f : X → Y , X scheme, in which amorphism between f : X → Y and f ′ : X ′→ Y is u : X → X ′ such that f ′ u= f .

Proposition 1.5.4

Let f : X → Y be a morphism of schemes. The following are equivalent :

1. The map f is affine.

2. There exists a covering Y =⋃

i∈IYi of Y such that ∀i ∈ I , f −1(Yi) is affine.

Proof. Omitted, see [14], p. 208-210

Example 1.5.5. A morphism between affine schemes is affine.

We will eventually introduce the notion of relative spectrum, which play for a quasi-coherent algebra A a similar role as Spec A for a ring A. This construction generalises thestatement of the theorem 1.2.6 : if A is a ring, B a A-algebra and X a A-scheme, there isa canonical bijection HomA(X , Spec B) ' HomA−algebras(B,OX (X )). Moreover, the categoriesof the A-algebras is equivalent to the category of affine schemes over Spec A ; here, thecategories of the quasi coherent OY -algebras and of the affine morphisms X → Y will beequivalent through the Spec functor.

Theorem 1.5.6

Let Y be a scheme and A be a quasi-coherent OY -algebra. We can find a Y -schemeSpecYA , with structural morphism f such that f is affine and f∗OSpecYA =A .

Moreover, if π : Z → Y is a Y -scheme,

ϕZ : HomY (Z , Spec YA ) −→ HomOY(A ,π∗OZ)

u 7−→ f∗u#

is a functorial bijection.

14

This means that if κ : W → Y is a Y -scheme and α : W → Z is a morphism ofY -schemes, then the following diagram commutes :

HomY (Z , Spec YA ) HomOY(A ,π∗OZ)

HomY (W, Spec YA ) HomOY(A ,κ∗OW )

ϕZ

ϕW

β 7→π∗α#βv 7→vα

See [14], page 444 for less informations ; another construction can be found in [4], p.41.Lemma 1.5.7 (Glueing schemes)

Let (X i)i∈I a family of schemes. If, for all i ∈ I , there exist a family (X i j) j∈I such that

X i j is an open subset of X i, and for all i, j ∈ I , an isomorphism of schemes fi j : X i j'→ X ji

such that :

1. fii = idX i

2. ∀i, j, k ∈ I , fi j(X i j ∩ X ik) = X ji ∩ X jk

3. (cocycle condition) The following diagram commutes :

X i j ∩ X ik Xki ∩ Xk j

X ji ∩ X jk

fik

f jkfi j

Then there exists a unique scheme X together with open embeddings ψi : X i → Xsuch that, for i, j ∈ I , ψi =ψ j fi j on X i j, and X =

⋃

iψ(X i).

Proof. Omitted, see [8], pp. 49-50.

Remark. If φi : X i → Y are Y -schemes, and the fi j is compatible with the φi, then theglueing X of the X i is a Y -scheme and its structural morphism φ satisfies, for every i ∈ I ,φ|X i= φi.

In the next proof, we will use some basic facts about the fibered products of schemesthat can be found in the third chapter [8] (pp. 78-85).

Notation : A commutative diagram of the form above is called cartesian if Z satisfies theuniversal property of the fibered product. We indicate it with the square in the center of thepicture.

Z X

X ′ Y

Proof (of 1.5.6). Let I be the set of open affine of Y . For i ∈ I , we denote by Ui the associatedopen affine. Let us set X i = SpecA (Ui), and fi : X i → Ui the morphism induced by thestructural morphism OY (Ui)→A (Ui). We define, for all j ∈ I , X i j = f −1

i (Ui ∩ U j).If Uk ⊂ Ui, we have the following cartesian diagram, because A (Uk) = A (Ui) ⊗OY (Ui)

OY (Uk) (it follows from [8], proposition 1.14 (b) p. 163 and proposition 1.12 (b) p. 162).

15

Xk X i

Uk Ui

αki

inc

fifk

(1.1)

As open embeddings are stable by base change, αki is an open embedding. Moreover,αki(Xk) = f −1

i (Uk). As αki is compatible with fk and fi, it is a morphism of Y -schemes.

Let i, j ∈ I . We want to define a family of isomorphisms βi j : X i j → X ji.Let us first notice that if k ∈ I is such that Uk ⊂ Ui ∩ U j, then the diagram 1.1 factors

through X i j – and likewise through X ji :

Xk X i j X i

Uk Ui ∩ U j Ui

αk,i, j

inc.

fi

αki

fk

It is then easy to see that the family of the Imαk,i, j when Uk ⊂ Ui ∩ U j is a covering ofX i j ; therefore, we can define βi j by glueing the morphisms :

X i j X ji

Xk

βi j

αk, j,iαk,i, j

under the condition that if Uk, Ul ⊂ Ui∩U j, (αk, j,iα−1k,i, j)| f −1

i (Uk∩Ul ) = (αl, j,iα−1l,i, j)| f −1

i (Uk∩Ul ),since the image of αk,i, j (resp. αl,i, j) is f −1

i (Uk) (resp. f −1i (Ul)).

Let us first consider the case where Ul ⊂ Uk. We have the following diagram :

X l Xk X i j

Ul Uk Ui ∩ U j

αlk αk,i, j

fl fk

the whole rectangle being cartesian. Therefore, by uniqueness of the fibered product, wehave αl,i, j = αk,i, j αlk, and likewise for X ji. Considering X l as an open subset of Xk, andpassing to the inverse, we get α−1

k,i, j| f −1i (Ul )

= α−1l,i, j and on the other hand αk, j,i|X l

= αl, j,i which

gives the result.

Now, in the general case, it suffices to show that for every Ur ⊂ Uk ∩ Ul , the maps

X i j X ji

X r

βi j

αk, j,i|Xrαk,i, j|Xr

and

X i j X ji

X r

βi j

αl, j,i|Xrαl,i, j|Xr

coincide.But according to the first case, αk,i, j|X r

= αr,i, j = αl,i, j|X r, with the same result for αr, j,i so

we have the wished equality.

16

The reason why the cocycle condition is true can be seen on this diagram (with Ui jk =Ui ∩ U j ∩ Uk) :

X v

f −1i (Ui jk) f −1

k (Ui jk)

X v f −1j (Ui jk) X vαv j

αvi

αvk

αvk

β jkβi j

βik

αv j

αvi

Furthermore, the structural morphism f : X → Y is obtained by glueing the X ifi→ Ui → Y ,

so f −1(Ui) = f −1i (Ui) = X i and f is affine. Likewise, for every i ∈ I , f∗OX (Ui) = OX i

(X i) =A (Ui) so as the open affine form a basis of open subsets of Y , f∗OX =A .

Let us prove the second part of the theorem. The commutativity of the diagram is clear.If π : Z → Y is a Y -scheme, we want to show that ϕZ is a bijection. Let β :A → π∗OZ

is a morphism ; we are looking for a u : Z → X = Spec YA such that f∗u# = β .

For every i ∈ I , β(Ui) yields, thanks to 1.2.6 a morphism υi : π−1(Ui)→ SpecA (Ui) =X i, which gives by composition with X i → X a morphism ui : π−1(Ui)→ X . Moreover, sincethe following diagram commutes, for Uk ⊂ Ui,

A (Ui) OZ(π−1(Ui))

A (Uk) OZ(π−1(Uk))

β(Ui)

β(Uk)

we have

π−1(Uk) Xk X

π−1(Ui) X i X

υk

υi

and thus ui|π−1(Uk) = uk, which, similarly to the construction of the βi j above, proves thatwe can glue the ui into a morphism of Y -schemes u : Z → X .

We defineψZ(β) = u. If u : Z → X is a morphism of Y -schemes, let us take β = ϕZ(u) =

f∗u# and u′ = (ψZ ϕZ)(u). We know that u′|π−1(Ui)

= π−1(Ui)υ′i−→ X i

inclusion−→ where υ′i is

induced by β(Ui). Moreover, as u is a morphism of Y -schemes, u(π−1(Ui)) ⊂ f −1(Ui) = X i

and u|π−1(Ui) can be decomposed as π−1(Ui)υi−→ X i

inclusion−→ X . Since u#(X i) = β(Ui), we have

υi = υ′i and finally u= u′.Furthermore, if β : A → π∗OZ is a morphism of OY -algebra, β ′ = (ϕZ ψZ)(β) is

actually β . Indeed, if u =ψZ(β), then for all i ∈ I , β ′(Ui) = u#( f −1(Ui)) = u#(X i) = β(Ui)by construction of u. Therefore, ϕZ and ψZ are inverses of each other.

Corollary 1.5.8

The categories QAlg(Y ) and Aff(Y ) are equivalent through

Aff(Y ) QAlg(Y )

f : X → Y f∗OX

Spec YA → Y AF

G

17

Proof. An affine morphism is quasi compact and separated (see [6], p. 321), so [8], p. 163implies that f∗OX is quasi-coherent.

If A ∈ QAlg(Y ), then (G F)(A ) = f∗OSpecYA = A , according to the theorem 1.5.6,where f : Spec YA → Y is the structural morphism.

If f : X → Y is an affine morphism, then X = Spec Y f∗OX . Indeed, with the samenotations as in the proof of 1.5.6, if i ∈ I , X i = Spec (OX ( f −1(Ui)) = f −1(Ui), since f isaffine. If i, j ∈ I , βi j : f −1(Ui ∩ U j)→ f −1(U j ∩ Ui) is the identity. Thus it is clear that X isthe glueing of the X i.

Let us study the action on the morphisms : if u : X → X ′ is a morphism between twoaffine morphisms f : X → Y and f ′ : X ′ → Y , then X = Spec YA , X ′ = Spec YA ′, whereA = f∗OX ,A ′ = f ′∗OX ′ and G(u) is given by the bijection ϕ : HomY (Spec YA ′, Spec YA )→HomOY

(A ,A ′) above. The diagram of the theorem 1.5.6 shows the functoriality.Likewise, the action of F an a morphism β : A ′ → A is given by the inverse of ϕ.

Hence, F and G are quasi inverses of each other for the morphisms.

1.6 Symmetric algebra and exterior powers of algebra

For proofs and details, see [6], pp. 196-199 (exterior powers) and 287-288 (symmetricalgebra).

Let A be a ring, and M be a A-module. We set ∀n ¾ 0, Tn(M) = M⊗n and T (M) =⊕

n¾0Tn(M). This module endowed with the product :

(m1 ⊗ · · · ⊗mn)× (m′1 ⊗ · · · ⊗m′l) = m1 ⊗ · · · ⊗mn ⊗m′1 ⊗ · · · ⊗m′lis a graded (non-commutative) A-algebra (i.e. ∀p, q ∈ N, Tp(M)Tq(M) ⊂ Tp+q(M)).

Definition 1.6.1

If I is the ideal of T (M) generated by the x ⊗ y − y ⊗ x , x , y ∈ M , then the symmetricalgebra of M is S(M) = T (M)/I . Endowed with the product induced by the product ofT (M), it is a commutative A-algebra.

As I is generated by homogeneous elements, we know that S(M) =⊕

n∈NSnM , where

SnM = Tn(M)/(I ∩ Tn(M)) = M⊗n/In, In being the submodule of M⊗n generated by them1 ⊗ · · · ⊗mn −mσ(1) ⊗ · · · ⊗mσ(n), σ permutation of 1, . . . , n.

The module SnM has the following universal property, that comes from the universalproperty of the quotient : it is endowed with a morphism π : M⊗n→ SnM , and if f : M⊗n→N , N A-module is a A-linear symmetric morphism (i.e. f (m1 ⊗ · · · ⊗mn) = f (mσ(1) ⊗ · · · ⊗mσ(n)) for every permutation σ), then f factors uniquely through SnM :

M⊗n N

SnM

f

efπ

A consequence of this property is that the construction of Sn· is functorial : if u : M → Nis a morphism, then we can build a morphism Snu : SnM → SnN such that

SnM SnN

M⊗n N⊗n

Snu

u⊗n

πM πN

18

Definition 1.6.2

If J is the ideal of T (M) generated by the x ⊗ x , x ∈ M , then the exterior algebra of Mis the A-module Λ(M) = T (M)/J .

As previously, Λ(M) =⊕

n∈NΛnM , where ΛnM = M⊗n/Jn, Jn = J ∩M⊗n being the submod-

ule generated by the m1 ⊗ · · · ⊗mn ∈ M⊗n such that ∃i 6= j : mi = m j. The module ΛnM iscalled the nth exterior power of M .

The universal property of ΛnM is that every alternating morphism f : M⊗n → N , NA-module, (i.e. ∃i 6= j : mi = m j ⇒ f (m1 ⊗ · · · ⊗mn) = 0) factors through ΛnM .

Again, Λn· acts also on the morphisms and therefore define a functor.

Notation : The class of m1⊗ · · ·⊗mn in ΛnM is denoted by m1∧ · · · ∧mn, while its classin SnM is denoted by m1 . . . mn.

Proposition 1.6.3

Let M be a free A-module of rank r and (e1, . . . , er) a basis of M .Then, for n ∈ N, the family of the (e j1∧· · ·∧e jr )J when J = j1 < · · ·< jr ⊂ 1, . . . , n

is a basis of the free A-module ΛnM .

In particular, if n= r, then Λr M is of rank 1, and is also called det M .

Proposition 1.6.4

Given n ∈ N, M a A-module, and p ∈ Spec A, (SnM)p = SnMp and (ΛnM)p = ΛnMp

Let us now generalize this construction to OX -modules, when X is a scheme.

Definition 1.6.5

Let F be a OX -module.

• The nth exterior powerΛnF ofF , is the sheafification of the presheaf U 7→ ΛnOX (U)F (U),

endowed with the restrictions

∀U ⊂ V ⊂ X , Λn(ρUV ) : ΛnOX (U)F (U) −→ Λn

OX (V )F (U)

s1 ∧ · · · ∧ sn 7−→ s1|V ∧ . . . sn|V

• SnF is the sheafification of U 7→ SnOX (U)F (U), endowed with the restriction SnρUV .

Once again, Λn· and Sn· are functors from the category of OX -modules into itself.

Proposition 1.6.6

Let F be a quasi-coherent OX -module. Then ΛnF and SnF are quasi-coherent. More-over, if U is an open affine of X andF|U = eM , then (ΛnF )|U =ßΛnM and (SnF )|U =ßSnM .

Remark. This proposition together with 1.6.4 shows that if x ∈ X , (SnF )x = SnFx and(ΛnF )x = ΛnFx .

19

Chapter 2

Double covers

2.1 Basic facts about covers

In this section, we fix a base scheme Y .

Definition 2.1.1

A d-cover of a scheme Y is an affine morphism f : X → Y , where f∗OX is a quasi-coherentlocally free OY -algebra of rank d. The number d is called the degree of the cover.

Definition 2.1.2

We denote, for d ∈ N∗, the category of the d-covers of Y by Covd(Y ).A morphism between f : X → Y and f ′ : X ′ → Y in this category is p : X → X ′

isomorphism of schemes such that

X X ′

Y

p

f ′f

Proposition 2.1.3

The restriction of the functors in 1.5.8 yields an equivalence of categories between thecategory of d-covers of Y and the category of quasi-coherent locally free OY -algebras ofdegree d, with isomorphisms as arrows.

Thus, considering a cover f : X → Y is the same as takingA ∈ QAlg(Y ) which is locallyfree. In this situation, f is the structural morphism SpecY (A )→ Y . For this reason, we willessentially consider the d-covers as locally free algebras of rank d in the next two chapters.

Example 2.1.4. • The 1-covers are the isomorphisms.

Indeed, if A is a quasi-coherent locally free algebra of rank 1, endowed with hisstructural morphism α : OY →A , then for all x ∈ Y , αx : OY,x →Ax is a morphism ofalgebras andAx is a OY,x -module of rank 1. As 1 /∈ mx , (1) is a basis of the k(x)-vectorspace of dimension 1Ax/mxAx . Thus, using Nayakama’s lemma, it is a generator ofAx . Thanks to the lemma 2.2.5, we know that (1) is a basis ofAx .

Therefore αx is an isomorphism since αx(1) = 1, and so is α.

• If K is a number field of degree d, then we know that the ring of the integers OK is afree Z-module of degree d. Therefore, SpecOK → SpecZ is a d-cover.

20

Example 2.1.5. We know that p : z→ z2 is a topological cover of C∗, but not of C becausep−1(0) has one element, whereas the cardinality of p−1(λ) is 2 if λ 6= 0. So 0 is a "doublepoint". We want to find an algebraic equivalent to p and see if the same phenomenon occursin 0.

Let k be a field, and consider the morphism of k-algebras ϕ : k[X ] −→ k[Z]X 7−→ Z2

. It

induces an affine morphism of schemes f : S→ T , where S = Speck[Z] and T = Speck[X ].

The quasi-coherent OT -module (for the structure given by f ) f∗OS is ßk[Z].But there is a surjective morphism of k[X ]-algebras α : k[X ][Y ] −→ k[Z]

Y 7−→ Z, whose

kernel is (Y 2−X ). Hence k[Z]' k[X ][Y ]/(Y 2−X ) = k[X ]⊕k[X ]k[X ]Y is a free k[X ]-moduleof rank 2 and f is a cover of degree 2.

Let us now replace k by C, or any algebraically closed field. We know by Hilbert’s Null-stellensatz that the maximal ideals of C[X ] are the (X − λ), λ ∈ C. Moreover (cf [8], p.83), if λ ∈ C, and p = (X − λ), f −1(p) is Sλ = S ×T Spec k(p), where k(p) = OT,p/mp 'C[X ]/(X −λ)' C is the residue field. Sλ satisfies the following base change diagram :

Sλ SpecC[Z]

Spec k(p) SpecC[X ]

f

So

Sλ = Spec

C[Z]⊗C[X ]C[X ](X −λ)

= Spec

C[Z](ϕ(X −λ))

= Spec

C[Z](Z2 −λ)

If λ 6= 0, and µ2 = λ, we deduce from the Chinese remainder theorem thatC[Z](Z2 −λ)

=

C[Z](Z −µ)

×C[Z](Z +µ)

' C×C, hence Sλ = SpecCt SpecC has two elements.

If λ = 0, S0 = Spec

C[Z](Z2)

= (Z) is a singleton, but it is the spectrum of a C-vector

space of dimension 2, which is a kind of multiplicity. We call this phenomenon ramification.

Proposition 2.1.6

LetA be a cover of degree d. There exists a unique morphism TrA :A →OY verifyingthe following property : for every U affine open subset of Y such that the OY (U)-moduleA (U) is free of rank d, TrA (U) is the standard trace application onA (U).

Moreover, TrA (1A ) = d.

Actually this theorem results from a more general principle that we describe in thislemma :Lemma 2.1.7

Let E be a locally free OY -module of rank d. There exists a unique morphism Tr :End(E )→OY verifying the following property : for every U affine open subset of Y suchthat the OY (U)-module E (U) is free of rank d, Tr (U) is the standard trace applicationEnd(E (U))→OY (U).

21

Proof. LetB be the set of the affine open subsets V of Y such that E|V =àE (V )' O dV . Since

E is locally free, B is a basis of open subsets of Y . We can therefore define a morphism ofOY -modules onB .

Let V ∈B , and λ be an isomorphism of OV -modules between E|V and O dV . The map λ is

entirely determined by the data of λ(V ) ∈ Iso(E (V ),OY (V )).We can define Tr λ,V on the free sheaf End(E )|V by

End(E|V ) End(O dV ) = Md(OV (V )) OV (V )

ϕλ: f 7→λ f λ−1

Ý

Tr

Let λ′ : E|V →O sV be another isomorphism. The diagram

End(E|V ) Md(OV (V ))

Md(OV (V )) OV (V )

ϕλ′

Ý

ϕλ′λ−1ϕλ

Ý

Tr

Tr

commutes because the ordinary trace map for matrices satisfies Tr (ABA−1) = Tr (B),∀A, B ∈Md(OV (V )), A invertible, and the diagonal map is an isomorphism. Hence Tr λ,V = Tr λ′,V .

Therefore, we can set Tr (V ) = Tr λ,V (V ) for every V ∈ B and λ : E|V ' O dV . In order

to see that Tr is well-defined as a map of sheaves, let us check, for W ⊂ V ∈ B , that thefollowing diagram is commutative :

End(E|V ) OY (U)

End(E|W ) OY (V )

Tr (V )

restriction

Tr (W )

restriction

Let us consider an isomorphism λ : E|V ' O dV . Then λ|W is an isomorphism between E|W

and O dW .

End(E|V ) Md(OV (V )) OY (V )

End(E|W ) Md(OW (W )) OY (W )

ϕλ Tr V

ϕλ|W

restrictionTr W

ρ restriction ψ

whereψ is the morphism of restriction coefficient by coefficient. Indeed, if M ∈ Md(OY (V )),then :

ψ(M) = (ϕλ|W ρ ϕλ−1)(M)

= λ|W (λ−1 M λ)|Wλ−1|W

= λ|W λ−1|W M|W λ|W λ−1

|W = M|W

But the restriction M|W as a morphism of sheaves correspond to the restricted matrix,since, if (e1, . . . , ed) is the canonical basis of OY (V ), then (e1|W , . . . , es|W ) is the canonical basisof OY (W ) and ∀i = 1 . . . d, M|W (ei|W ) = M(ei)|W = (

∑

j mi, je j)|W =∑

j mi, j|W e j|W .

With thisψ, it is clear that the diagram from the right commutes. Thus, the big rectanglecommutes. As Tr (W ) doesn’t depend on the isomorphism we choose, the horizontal arrows

22

correspond to Tr (V ) and Tr (W ). Hence, Tr is well defined on B and coincides with theactual trace map when E|V is free.

Proof (of 2.1.6). Let ϕ : A −→ End(A )a ∈A (U) 7−→ (ta : a′ 7→ a ·A (U) a′)

Composing ϕ with the trace map built in the lemma gives us a morphism TrA :A →OY

that satisfies the conditions of the theorem.Moreover, TrA (1A ) = d since they coincide locally : if U is an open affine such that

A (U) is free of rank d, TrA (U)(1A|U) = d.

Remark. IfA is a locally free OY -algebra of rank d, then for every x ∈ Y , (TrA )x = TrAx/OY,x.

Proposition 2.1.8

If Fu−→G

v−→F are morphisms of sheaves such that v u= idF , then if j : ker v→G

is the inclusion, u⊕ j :F ⊕ker v→G is an isomorphism and its inverse is v⊕ (id−uv) :G →F ⊕ ker v.

In particular, ifA is a d-cover and d ∈ OY (Y )∗, thenA ' OY ⊕ ker TrA .

Proof. A direct computation show the first assertion.

Here,TrA

d f = id.

OY A OYTrA /df

Therefore, β = f ⊕ inclusion : OY ⊕ kerTrA →A is an isomorphism.

Remark. The first statement is not specific to the sheaves, it is also true for modules forinstance.

◊ ◊ ◊ ◊

Our goal in the next section and the next chapter is to find a concrete description ofCov2(Y ) and Cov3(Y ). Essentially, given (A ,m) a locally free algebra of rank 2 or 3, we willtry to find the "smallest" component of the multiplication m that contain all the information.

2.2 Double covers via line bundles

Let Y be a scheme with 2 ∈ OY (Y )∗.

Let D2(Y ) the category whose objects are the pairs (L ,σ) where L is an invertiblesheaf on Y and σ :L ⊗L → OY is a morphism of OY -modules. In this category, a morphismbetween (L ,σ) and (L ′,σ′) is an isomorphism p :L →L ′ such that

L ⊗L L ′ ⊗L ′

OY

σ σ′

p⊗p

23

Consider he following associations :

• Λ from Cov2(Y ) to D2(Y ) which associates with a locally free algebra (A ,m) the

object (kerTrA ,σ), where σ :L ⊗Linclusion−→ A ⊗A

m−→A

TrA /2−→ OY .

If u :A →A ′ is an isomorphism of OY -algebras, wet Λ(u) := u|ker TrA .

• ∆ fromD2(Y ) to Cov2(Y ) that associates with (L ,σ) the OY -algebraL⊕OY , endowedwith the multiplication m : (OY ⊗OY )⊕ (OY ⊗L )⊕ (L ⊗OY )⊗ (L ⊗L ) −→ OY ⊕Lwhose last component is σ ⊕ 0 : L ⊗L → OY ⊕L and the others are the naturalones following from the desired compatibility with the OY -module structure. Moreprecisely, for every open subset U :

m(U)((a⊕ l)⊗ (a′ ⊕ l ′)) = (aa′ +σ(U)(l ⊗ l ′))⊕ (al ′ + a′l)

If v :L →L ′ is a morphism in D2(Y ), we set ∆(v) = v ⊕ id.

The main result of this chapter is :

Theorem 2.2.1

The functors Λ and ∆ are well-define and quasi-inverses of each other. Thus, Cov2(Y )and D2(Y ) are equivalent categories.

We will see that we can actually study a "local" simplified problem corresponding tothe situation we get if we localize the global problem by interesting ourselves to the stalks.Solving the local case is only a matter of commutative algebra. For the double covers as wellas the triple ones, the only difficulty in the global case is therefore to define global mapsthat we can localize to use the local case.

Λ is well-defined

Proposition 2.2.2

If E is a locally free OY -module of finite rank, and E1,E2 quasi-coherent OY -modulessuch that E = E1 ⊕E2, then E1,E2 are locally free of finite rank.

We admit the following lemma :

Lemma 2.2.3

If F is a quasi-coherent OY -module and Y =⋃

i∈IUi, where Ui is open and affine, then F

is finitely presented (i.e. for every U open affine, F (U) is finitely presented) if and onlyif for every i ∈ I , F (Ui) is finitely presented.

Proof (of the proposition 2.2.2). Let us first prove that E1 and E2 are finitely presented.Let Y =

⋃

i∈IUi be an open covering of Y such that E|Ui

' O niY |Ui

. We have

0 E2 E E1 0projectioninjection

And

E2 E

E

injection

idproj

24

so that E → E → E1 → 0, and by restricting this exact sequence to Ui we get a finitepresentation of E1|Ui

. The lemma ensures us that E1 is finitely presented. Symetrically, it’salso the case for E2.

According to the proposition 1.4.3, we know that, for i = 1, 2, Ei is locally free of finiterank if and only if ∀x ∈ Y , Ei,x is a free OY,x -module of finite rank. We are brought back toa local situation that is solved in the proposition 2.2.4.

Proposition 2.2.4

If (R,m) is a local ring and M a free R-module of finite rank such that M splits inM = N ⊕ L, N , L R-modules. Then N and L are free of finite rank.

Lemma 2.2.5

If (R,m) is a local ring, a surjective morphism between two free R-modules of samefinite rank is injective.

Proof. Let ϕ : Rn→ Rn be such a surjective morphism and K = kerϕ.Let ψ a morphism such that ϕ ψ = id. The proposition 2.1.8 gives an isomorphism

ψ : Rn → Rn ⊕ K , which pass to the quotient in ψ : kn '−→ kn ⊕ K/mK , where k = R/m.

Hence K/mK = 0. Moreover, K is of finite type since the projection Rn ' Rn ⊕ K → K issurjective, so by Nakayama’s lemma, K = 0.

Proof (of the proposition). To simplify, we can assume that M = Rn. Let us tensorise by R/m: if k = R/m, kn = N/mN ⊕ L/mL as k-vector spaces. Let s = dimk N/mN , then L/mL is ofdimension n− s.

By Nakayama’s lemma, if (x1, . . . , xs) is a basis of N/mN , then ϕ1 : Rs −→ Nei 7−→ x i

is

surjective, where (ei) is the canonical basis of Rs.Symetrically, we have ϕ2 : Rn−s L and thus Rs ⊕ Rn−s N ⊕ L = Rn. hence, according

to the lemma 2.2.4, this morphism is injective and so are ϕ1 and ϕ2.

Here, the proposition 2.1.8 states thatA = ker TrA ⊕OY and kerTrA is quasi-coherentsince it is the kernel of a morphism between quasi-coherent sheaves (cf [8], p. 162). AsAis locally free of finite rank, so is kerTrA and passing to the stalks, we see it is of degree 1,i.e. invertible.

If u is a morphism between A and A ′ in Cov2(Y ), then we have to check that v =Λ(u) = u|ker TrA is an isomorphism onto kerTrA ′ , and preserves σ,σ′.

These three statements can be checked locally, i.e. on the stalks. So let us consider thissituation : R is a local ring where 2 ∈ R∗, A= kerTr A⊕ R and A′ = ker Tr ′A⊕ R are two freeR-algebras of rank 2, σ and σ′ are defined as in the global case, and u is an isomorphism ofR-algebras between A and A′. We set v = u|kerTr A

.

First, let (ei)i be a basis of A, and a ∈ A. Then (u(ei))i is a basis of A′, and tu(a) : b′ 7→u(a)b′ sends u(ei) to u(aei). Thus, the matrix of tu(a) in (u(ei)) is the same as the one of ta

in ei and Tr A′(u(a)) = Tr A(a). This proves the first fact.On the other hand, v is an isomorphism because it is injective and if a′ ∈ ker Tr A′ , there

exists a ∈ A such that a′ = u(a), so 0= Tr (a′) = Tr (a) and a ∈ kerTr A.

25

Finally, the compatibility of v with σ,σ′ results from the fact that u is a morphism ofR-algebras, combined with Tr A′ u= Tr A.

∆ is well-defined

We want to prove two things : first that m is commutative and associative. We noticedin the section 1.5 that this was equivalent to the commutativity and associativity of the mapon the stalk mx for every x ∈ Y . Furthermore, if v :L →L ′ is a morphism of OY -modulesand u = v ⊕ id : L ⊕ OY → L ′ ⊕ OY , we want u to be a morphism of OY -algebras. This istrue if and only if, for all x ∈ Y the map on the stalks ux is a morphism of OY,x -algebra

That’s why we can assume that Y = Spec R, with R local ring. We are brought back tothe study of (L,σ), where L is a free R-module of rank 1, (R,m) being a local ring in which2 is invertible, and σ : L ⊗ L → R is a morphism of R-modules. The multiplication map onA= L ⊕ R we are interested in is m induced by σ⊕ 0.

Lemma 2.2.6

Let z be a generator of L and σ = σ(z⊗z) Then A' R[X ]/(X 2− σ) as a R-module, withrespect to the multiplications.

Proof. Let ψ : R[X ]/(X 2 − σ) = R⊕ Rx −→ A= R⊕ RzX = x 7−→ z

, defined as a R-linear map. ψ

is clearly bijective. Moreover, it preserves the multiplication, since :

ψ((a+ bx)(c + d x)) =ψ(ac + bdσ+ (ad + bc)x) = ac + bdσ+ (ad + bc)z

And m(ψ(a+ bx)⊗ψ(c + d x)) =m((a+ bz)⊗ (c + dz)) = ac + bdσ+ adz + bcz.

So the map m has the same properties as the multiplication on R[X ]/(X 2 − σ) andtherefore is commutative and associative.

Let v be an isomorphism L→ L′ that preserve σ,σ′. We want to show that u= v⊕ id isa morphism of R-algebras and an isomorphism.

u can be seen as a morphism from R[X ]/(X 2− σ) = R⊕Rx to R[X ]/(X 2− σ′) = R⊕Rx ′.Moreover, the compatibility with σ,σ′ means that u(x)2 = σ(x ⊗ x) = σ.

So u((a+ bx)(c + d x)) = u(ac + bdσ+ (ad + bc)x) = ac + bdσ+ (ad + bc)u(x), and

u(a+ bx)× u(c + d x) = (a+ bu(x))(c + du(x)) = ac + bdu(x)2 + (ad + bc)u(x)= ac + bdσ+ (ad + bc)u(x)

Thus u is indeed a morphism of R-algebras, and obviously an isomorphism.

∆ and Λ are quasi inverses of each other.

1. ∆ Λ' 1Cov2(Y).

Let (A ,m) be a locally free OY -algebra of rank 2, (L ,σ) = Λ(A ,m) (so L =kerTrA ,m).

Let us denote the multiplication induced by σ onA =L ⊕OY by n. We want to showthat m= n, which can be proved locally.

26

So let us take these notations : (R,m) is a local ring such that 2 ∈ R∗ (A,m) is afree R-algebra of rank 2, L = ker Tr A,m is endowed with σ induced by m, n is themultiplication induced by σ on A.

We know that A= L ⊕ R = Rz ⊕ R, and (A,n) ' R[X ]/(X 2 − σ), σ = σ(z ⊗ z). We setm(z ⊗ z) = a+ bz, a, b ∈ R.

Therefore 0= Tr A,m(z) = Tr

0 a1 b

= b.

Moreover, Tr A,m(m(z⊗ z)) = 2a. But σ =Tr (A,m)

2m

|L⊗L, so actually, a = σ, which

means that m(z ⊗ z) = σ(z ⊗ z) = n(z ⊗ z). Thus, m= n.

Finally, if u is a morphism from (A ,m) to (A ′,m′), then (∆ Λ)(u) = u|L ⊕ id,L = kerTrA ,m. The decomposition A = OY ⊕L combined with the fact that u is amorphism of OY -algebras implies that (∆ Λ)(u) = u.

2. Λ ∆' 1D2(Y).

Let us consider (L ,σ) ∈ D2(Y ). If ∆(L ,σ) = (A ,m), then (Λ ∆)(L ,σ) = (M ,ν),whereM = kerTrA ,m, and ν is induced by m.

Once again, the equality L =M and σ = ν can be checked on the stalks, so we goback to the local situation as above.

We have A = L ⊕ R = Rz ⊕ R and we know that A ' R[X ]/(X 2 − σ) as R-algebras,σ = σ(z ⊗ z).

Tr A,m(z) = Tr

0 σ1 0

= 0 (matrix in the basis (1, z)), so L ⊂ M .

Moreover, if w is a basis of M , then there exists a, b ∈ R such that w = a + bz and

0= Tr (w) = Tr

a bσb a

= 2a, hence a = 0 because 2 ∈ R∗. So w ∈ L and L = M .

Finally, if l, l ′ ∈ L, σ(l⊗l ′) ∈ R, soσ(l⊗l ′) =Tr2(σ(l⊗l ′)) =

Tr2(m((0⊕l)⊗(0⊕l ′))) =

ν(l ⊗ l ′) by construction, because M = L.

If v is a morphism between (L ,σ) and (L ′,σ′), then Λ(∆(v)) = (v ⊕ id)|kerTrL⊕OY=

(v ⊕ id)|L according to the computation above, so Λ(∆(v)) = v.

27

Chapter 3

Triple covers

In this chapter, we will mainly refer to the article of Rick Miranda ([9]).

Let us first describe the framework of the construction. We consider in all this part Y ascheme such that 6 ∈ OY (Y )∗.

LetD3(Y ) be the category whose objects are pairs (E ,δ)where E is a locally free sheaf ofrank 2 on Y and δ : S3E→ Λ2E is a morphism of OY -modules. In this category, a morphismbetween (E ,δ) and (E ′,δ′) is an isomorphism u : E → E ′ such that

S3E S3E ′

Λ2E Λ2E ′

S3u

δ δ′

Λ2u

commutes.Consider the following associations :

• If (A ,m) is a locally free OY -algebra of rank 3, then thanks to the proposition 2.1.8(3 ∈ OY (Y )∗), we have A = E ⊕ OY , E = kerTrA , and we can define β : S2E → E as

the factorisation through S2E of E ⊗ Em|E⊗E−→ E ⊕ OY

projection−→ E . We will prove in 3.2.2

that there exists a unique δ : S3E → detE such that this diagam commutes :

S2E ⊗ E E ⊗ E

S2E detE

β⊗id

δ

(3.1)

We define H from Cov3(Y ) to D3(Y ) which associates with (A ,m) the object (E ,δ).

If u :A →A ′ is an isomorphism of OY -algebras, we set H(u) = u|E .

• If (E ,δ) is an object of D3(Y ), then thanks to the proposition 3.2.5, we can find aunique β : S2E → E satisfying the diagram 3.1. Moreover, we will show in the propo-sition 3.2.3 that there exists a unique η : S2E → OY =A such thatA endowed withm induced by β ⊕ η is a OY -algebra. We set K(E ,δ) = (A ,m) and for v : E → E ′morphism in D3(Y ), K(v) = v ⊕ id.

The main result of this chapter is :

28

Theorem 3.0.7

The functors K and H are quasi inverses of each other and therefore D3(Y ) and Cov3(Y )are equivalent categories.

To make the proof clearer, we add a third category that will be equivalent to the firstones : the objects of eD3(Y ) are the pairs (E ,β), where E is a locally free sheaf of rank 2and β : S2E → E is a morphism of OY -modules such that there exists a unique morphism δmaking the diagram 3.1 commutative.

3.1 The local case

Let (R,m) be a local ring such that 6 ∈ R∗.

Let LCov3(R) be the set of the free algebras (A,m) of rank 3. We know that if A ∈LCov3(R), then A= ker Tr A⊕R and kerTr A is a free R-module of rank 2 (theorem 2.1.8 andproposition 2.2.4).

Let d3(R) be the set of the pairs (E,φ2) where E is a free R-module of rank 2 and φ2

is a morphism from S2E to E such that, if (z, w) is a basis of E, the matrix of φ2 in the

basis (z2, zw, w2) of S2E is of the form

a −d cb −a d

, (a, b, c, d) ∈ R4. This condition doesn’t

depend on the basis we choose, thanks to the following lemma, which also shows that d3(R)is the local equivalent of D3(Y ).Lemma 3.1.1

Let R be a ring, E a free R-module of rank 2 and β : S2E→ E a morphism. If (z, w) is abasis of E, then the existence of the factorisation

S2E ⊗ E E ⊗ E

S3E det E

β⊗id

δ

νµ

is equivalent to : M(z2,zw,w2)(β) =

a −d cb −a d

where a, b, c, d ∈ R.

Proof. A direct check shows that the existence of the factorisation is equivalent to kerµ ⊂kerψ, with ψ= ν (β ⊗ id).

But the kernel of µ is generated by (z2 ⊗w− zw⊗ z, zw⊗w−w2 ⊗ z).

Moreover, if M(z2,zw,w2)(β) =

a e cb f d

, a direct computation shows that ψ(z2 ⊗ w −

zw⊗ z) = (a+ f )z ∧w and ψ(zw⊗w−w2 ⊗ z) = (d + e)z ∧w.So

kerµ ⊂ kerψ⇔§

a = − fe = −d ⇔ M(z2,zw,w2)(β) =

a −d cb −a d

We are going to explain why taking a free R-algebra of rank 3 is the same as consideringan object (E,φ2) of d3(R).

29

Let (A,m) ∈ LCov3(R), A = E ⊕ R, where E = kerTr A. As m is commutative, we get φsuch that :

E ⊗ E A

S2E

m|E⊗E

φ

And we define φ = (φ1 : S2E→ R)+(φ2 : S2E→ E). For the same reasons as in the caseof the double covers, the interesting part of m : (R⊗R)⊕ (R⊗ E)⊕ (E⊗R)⊕ (E⊗ E)→ E⊕Ris expressed by φ, since the other components are imposed by the fact that A is a R-algebra.

Let (z, w) be a basis of E. We can find (a, b, c, d, e, f , g, h, i) ∈ R9 such that

φ(z2) = g + az + bwφ(zw) = h+ ez + f wφ(w2) = i + cz + dw

m is associative so in particular m(z⊗m(z⊗w)⊗w) =m(m(z⊗ z)⊗w) (condition (1))and m(z ⊗m(w⊗w)) =m(m(z ⊗w)⊗w) (condition (2)).

Using the expression of φ, and the compatibility of m with the structure of R-module,we get for (1) gw+ a(h+ ez+ f w)+ b(i+ cz+ dw) = hz+ e(g + az+ bw)+ f (h+ ez+ f w),i.e.

(ah+ bi) + (ae+ bc)z + (a f + bd + g)w= (eg + f h) + (h+ ae+ e f )z + (be+ f 2)w

and for (2) :

(cg + dh) + (i + ac + de)z + (bc + d f )w= (eh+ f i) + (e2 + c f )z + (h+ e f + d f )w

As (1, z, w) is a basis of A, we finally obtain :

g = f 2 + be− bd − a fh= bc − e f

i = e2 + c f − ac − de(3.2)

In other words, φ1 is determined by φ2.

Moreover, E = ker Tr A so 0 = Tr (z) = Tr

0 g h1 a e0 b f

= a+ f and likewise 0 = Tr (w) =

e+ d. So

M(z2,zw,w2)(φ2) =

a −d cb −a d

(3.3)

i.e. (E,φ2) is in d3(R). Furthermore :

M(z2,zw,w2)(φ1) =

2(a2 − bd) −(ad − bc) 2(d2 − ac)

(3.4)

We have reduced the number of parameters of the problem from 9 to 4.

30

Reciprocally, let us take (E,φ2) ∈ d3(R). If (z, w) is a basis of E, the matrix of φ2 in the

basis (z2, zw, w2) of S2E is φ2 =

a −d cb −a d

. We can define

φ1 =

2(a2 − bd) −(ad − bc) 2(d2 − ac)

morphism from S2E to R ⊂ A= E ⊕ R, andthus φ : S2E→ A defining a multiplication map n on A.

n is associative, since it satisfies the conditions 3.2 which are clearly sufficient.Furthermore :

E = ker Tr A,n (3.5)

because Tr A,n(z) = Tr

0 2(a2 − bd) −(ad − bc)1 a −d0 b −a

= 0 and likewise Tr A,n(w) = 0 ;

moreover, if x = p + qz + rw ∈ A is such that Tr A,n(x) = 0, then 3p = 0, hence p = 0 since3 is invertible in R, and x ∈ E.

In conclusion, the way we have constructedφ2 out of m and n out ofφ2 allows us clearlyto state the following result :

Theorem 3.1.2

The two maps we have built, ∆ : d3(R) −→ LCov3(R)(E,φ2) 7−→ (E ⊕ R,n)

and Λ : LCov3(R) −→ d3(R)(A,m) 7−→ (E,φ2)

are inverses of each other.

Proof. The fact that (∆Λ)(A,m) = (A,m) is obvious by construction, and (Λ∆)(E,φ2) =(E,φ2) principally because if n is the multiplication induced byφ2, then E = kerTr E⊕R,n.

3.2 The global case

Let Y be a scheme such that 6 ∈ OY (Y )∗.

3.2.1 Equivalence between Cov3(Y ) and eD3(Y )Theorem 3.2.1

There is an equivalence of categories between Cov3(Y ) and eD3(Y ) given by :

Cov3(Y ) eD3(Y )

(A ,m) (E = kerTrA ,β)

(E ⊕OY ,n) (E ,β)Ω

Θ

where in the first line, β = E ⊗ Em|E⊗E−→ E ⊕OY

projection−→ E .

31

Θ is well-definedProposition 3.2.2

GivenA = OY ⊕E , locally free OY -algebra of rank 3, with E = kerTr A, whose multipli-cation induces β : S2E → E , there exists a unique δ : S3E → detE such that

S2E ⊗ E E ⊗ E

S3E detE

β⊗id

δ

νµ

Proof (of the proposition 3.2.2). As in 3.1.1, we immediately see that the existence of thefactorisation is equivalent to kerµ ⊂ kerψwhereψ= ν(β⊗id). This is the same as sayingthat kerµx ⊂ kerψx for all x ∈ Y , that is to say (Ex ,βx) ∈ d3(OY,x) which is true. Indeed βx

corresponds in the local case, with R = OY,x to the map φ2 and the formula 3.3 shows theresult.

Construction of Ω

Let (E ,β) be an object of eD3(Y ).Proposition 3.2.3

There exists a unique η : S2E → OY such that β ⊕ η : S2E → OY ⊕ E = A induces astructure of OY -algebra.

Proof. We will proceed in two steps : first, proving the theorem when Y = Spec (R) is affineand E = eE, E free R-module of rank 2. Then we will prove the general case.

First step : Let us fix (z, w) basis of E. As in the proof of 3.2.2, the matrix of β in

(z2, zw, w2) is of the form

a −d cb −a d

.

• Existence : Keeping in mind the local computation leading to 3.4, we define η by thematrix

2(a2 − bd) −(ad − bc) 2(d2 − ac)

and we know that, as the stalks of β⊕ηsatisfies the conditions 3.2, the induced multiplication is associative.

• Uniqueness : If η′ : S2E → OY is such that φ′ = β ⊕ η′ induces an associative mul-tiplication map on E ⊕ OY , then by looking at the stalks and using the computationleading to the system 3.2, we show that the matrix of η′ is necessarily

2(a2 − bd) −(ad − bc) 2(d2 − ac)

, whence η= η′.

Second step : LetB be the basis of open subsets of Y composed by the U open affine ofY such that E|U is free of rank 2. Thanks to the first step, for U ∈B , we have built a uniqueηU corresponding to βU = β|U .

Now, if V ⊂ U are two elements of B , the multiplication induced by βU ⊕ηU restrictedto V induces a structure of OY -algebra on E|V .

In other terms, (ηU)|V satisfies the property characterising ηV , which implies that theyare equal. In particular, this result gives the compatibility with the restriction maps whichallows us to define a morphism η of sheaves by setting ∀U ∈B ,η(U) = ηU(U).

The uniqueness in the general case ensues from the uniqueness in the first step becauseif η′ is a morphism such that η⊕ β yields a structure of OY -algebra on E , then for U ∈ B ,η′|U ⊕ β|U induces a structure of OY -algebra on E|U and thus η′|U = η|U . Hence η′ = η.

32

Proposition 3.2.4

If u : (E ,β)→ (E ′,β ′) is a morphism in eD3(Y ), then v = Ω(u) = u⊕ id : E ⊕OY →E ′⊕OY

is an isomorphism of OY -algebras.

Proof. Let us take ηβ ,ηβ ′ as in proposition 3.2.3. If v = id⊕u= Ω(v), then, as u is compat-ible with β ,β ′, we only have to prove :

S2E S2E ′

OY OY

S2u

id

ηβ′ηβ

To show that η := ηβ ′ S2u is equal to ηβ , we can demonstrate that η ⊕ β induces astructure of OY -algebra onA and use the uniqueness in the proposition 3.2.3.

As v is a isomorphism, we can carry the multiplication m′ ofA ′ toA by

A ⊗A A ′ ⊗A ′

A A ′

v⊗v

v−1

em m′

But, if n is the "multiplication" map induced by η⊕ β onA , we have

E ⊗ E E ′ ⊗E ′

S2E S2E ′

OY ⊕E OY ⊕E ′

m′E ′⊗E′S2u

id⊕u

η⊕β ηβ′⊕β

nE⊗E

u⊗u

The previous diagram implies that n= em, so n induces a structure of OY -algebra onA .Therefore, by uniqueness, ηβ = η.

Ω and Θ are quasi inverses of each other

For the morphisms, the situation is the same as for the double covers.For the objects :

• If (A ,m) ∈ Cov3(Y ), then (ΩΘ)(A ,m) = (E⊕OY ,n), where E = ker Tr (A ,m) and n isinduced by the component β : E ⊗E → E of m according to E . We know (proposition3.2.3) that the essential part of n is β ⊕ η, where, if in a local basis, M(z2,zw,w2)(β) =

a −d cb −a d

, η is of the form

2(a2 − bd) −(ad − bc) 2(d2 − ac)

. But according

to the formula 3.4, the local matrix of the essential component of m is the same, som= n.

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• If (E ,β) ∈ eD3(Y ), then taking Ω(E ,β) = (A ,n) with A = E ⊕ OY , we have (Θ Ω)(E ,β) = (ker Tr (A ,n),β

′). But thanks to the equation 3.5, E = ker Tr (A ,n) and itfollows obviously that β = β ′.

3.2.2 Equivalence between eD3(Y ) and D3(Y )

Let us define two functors F : eD3(Y )→D3(Y ) and G : D3(Y )→ eD3(Y ).

• F(E ,β) = (E ,δ) where δ is the unique morphism making this diagram commutative

S2E ⊗ E E ⊗ E

S3E detE

β⊗id

δ

νµ

• G(E ,δ) = (E ,β) where β is the unique morphism : S2E → E such that this diagramcommutes :

S2E ⊗ E E ⊗ E

S3E detE

β⊗id

δ

νµ

(3.6)

Our goal is to show that F and G are quasi inverses of each other ; obviously, only thefact that they are well-defined matters.

The theorem 1.4.7 will be the key argument to prove this equivalence of categories.

Action on the objects

As the fact that F(E ,β) is well-defined is ensured by the definition of eD3(Y ), we justhave to prove the following proposition to show that what we wrote makes sense.

Proposition 3.2.5

Given (E ,δ) ∈ D3(Y ), there exists a unique morphism β such that the diagram 3.6commutes.

Lemma 3.2.6

If E is locally free of rank 2, there exists a canonical isomorphism γ : E'→ detE ⊗ E∨.

Proof. Let ν : E ⊗E → detE be the canonical morphism. Then, thanks to the theorem 1.4.7,we get a morphism γ : E → detE ⊗ E∨.

If x ∈ Y , let us check that γx is an isomorphism. We can take a basis (v, w) of Ex .By remembering the construction of γ, one sees that if u : OY,x → Ex ⊗ (Ex)∨ is the

morphism as in the proof of 1.4.7, then

γx : vid⊗u7−→ v ⊗ (v ⊗ v∗ +w⊗w∗)

ν⊗id7−→ (v ∧ v)⊗ v∗ + (v ∧w)⊗w∗ = (v ∧w)⊗w∗

And similarly γx(w) = −(v ∨w)⊗ v∗, which proves that γx sends a basis on a basis, andthus is an isomorphism.

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Proof (of the proposition 3.2.5). We want to prove that, if δ : S3E → detE is a morphism,there exists a unique β : S2E → E such that

S2E ⊗ E E ⊗E

S3E detE

β⊗id

δ

νµ

If ϕ = δ µ : S2E ⊗E → detE , then the theorem 1.4.7 gives us a morphism ψ= K(ϕ) :

S2E → detE ⊗ E∨, where K is the bijection Hom(S2E ⊗E , detE )K−→ Hom(S2E , detE ⊗ E∨)

Let us take β = γ−1 ψ, with γ the map of the lemma 3.2.6. We want ϕ = ν (β ⊗ id) = λ,i.e. λ= K−1(ψ).

But, if ev is the evaluation morphism ζ⊗ s 7→ ζ(s), K−1(λ) is

S2E ⊗ Eψ⊗id−→ detE ⊗ E∨ ⊗E

iddetE⊗ev−→ detE

And λ can be decomposed as

S2E ⊗ Eψ⊗id−→ detE ⊗ E∨ ⊗E

γ−1⊗id−→ E ⊗E

ν−→ detE

Therefore it suffices to show that ν (γ−1 ⊗ id) = iddetE ⊗ ev. Passing to the stalks Ex

and using the computation made in the proof of the previous lemma gives us this results byevaluating on a basis.

Action on the morphisms

Proposition 3.2.7

If u ∈ HomeD3(Y )((E ,β), (E ′,β ′)), and δ,δ′ are the images of β ,β ′ by F , then the following

diagram commutes :

S3E S3E ′

detE detE ′

S3u

det u

δ′δ

Proof. As the morphisms νE : E ⊗ E → detE and µE : S2E ⊗ E → S3E are canonical, weobviously have the following diagrams :

E ⊗ E E ′ ⊗E ′

detE detE ′

u⊗u

νE νE′

det u

and

S2E ⊗ E S2E ′ ⊗E ′

S3E S3E ′

S2u⊗u

S3u

µE′µE

But we also have, by taking the tensor product of the compatibility diagram of u withβ ,β ′ and u :

S2E ⊗ E S2E ′ ⊗E ′

E ⊗ E E ′ ⊗E ′

S2u⊗u

β⊗id β ′⊗idu⊗u

Therefore :

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S2E ⊗ E S2E ′ ⊗E ′

E ⊗ E E ′ ⊗E ′

detE detE ′

S2u⊗u

det u

u⊗u

νE

ψ′

νE′

β⊗id

ψ

β ′⊗id

where the composed arrow ψ (resp. ψ′) is, by property of δ and β equal to ϕ = δ µE(resp. to ϕ′ = δ′ µE ′). This implies that in the following diagram, the whole rectanglecommutes.

SE ⊗E S2E ′ ⊗E ′

S3E S3E ′

detE detE ′

S2u⊗u

det u

S3u

δ

ϕ′

δ

µE

ϕ

µE

As we already know that the upper diagram commutes and the vertical arrows of thisdiagram are surjective (this principle is easy to understand set-theoretically, and we get thisresult here by looking at the stalks), then the lower diagram commutes.

Proposition 3.2.8

If u ∈ HomD3(Y )((E ,δ), (E ′,δ′)), and β ,β ′ are the images of δ,δ′ by G, then the followingdiagram commutes :

S2E S2E ′

E E ′

S2u

β β ′

u

Proof. We keep the notations of the previous proof. If ϕ = δµE , ϕ′ = δ′µE ′ , then glueingthe compatibility diagram of u with δ,δ′ and the canonical diagram of µE ,µE ′ , we get :

S2E ⊗ E S2E ′ ⊗E ′

detE detE ′

S2u⊗u

det uϕ′ϕ

Tensorising this diagram with E∨, (E ′)∨, where (u−1)∨ : f 7→ f u−1 is the transpositionof u−1, we have :

S2E ⊗ E ⊗E∨ S2E ′ ⊗E ′ ⊗ (E ′)∨

detE ⊗ E∨ detE ′ ⊗ (E ′)∨

S2u⊗u⊗(u−1)∨

det u⊗(u−1)∨ϕ′⊗idϕ⊗id

Finally, using the canonical morphisms τ : OY → E ⊗ E∨ and γ : E → detE ⊗ E∨ (con-structed respectively in 1.4.7 and 3.2.6), we have :

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S2E ⊗OY S2E ′ ⊗OY

S2E ⊗ E ⊗E∨ S2E ′ ⊗E ′ ⊗ (E ′)∨

detE ⊗ E∨ detE ′ ⊗ (E ′)∨

E E ′γ′−1γ−1

S2u

id⊗τ′S2u⊗u⊗(u−1)∨

ϕ⊗id ϕ′⊗id

id⊗τ

det u⊗(u−1)∨

u

By construction of the β ,β ′ corresponding to δ,δ′ (proposition 3.2.5), the vertical ar-rows are actually β and β ′.

These two propositions show that both functors are well-defined for the morphisms,which completes the proof of the equivalence of categories.

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