TWISTED FORMS IN GEOMETRY AND ALGEBRA EOIN MACKALL We study objects defined over a base field k, and see how these objects simplify as we move over extensions of k. This is akin to how a polynomial in one variable may be irreducible over a given field but, over certain field extensions the polynomial splits into smaller pieces, eventually to linear terms (and over an algebraic closure, every polynomial completely splits). In the language below, we’d call an irreducible polynomial a twisted form of a product of linear terms. After recalling some notions from algebraic geometry and cohomology in Section 1, we’ll move to the study of Severi-Brauer varieties in Section 2. Severi-Brauer varieties are twisted forms of projective space. This means Severi-Brauer varieties are varieties that become isomorphic to projective space after moving to a finite extension of the base field. As such, they have interesting rationality properties and serve as useful nontrivial examples in algebraic geometry. In Section 3 we carry out a study, similar to that of Section 2 but, with the objects of interest being central simple algebras. Central simple algebras are twisted forms of matrix algebras. This means central simple algebras are associative (but not necessarily commutative) k-algebras which become isomorphic to a matrix algebra over a finite extension of their base field. Due to Morita theory this takes on the particularly simple form of the study of division algebras over k (however, it remains a complicated problem to classify division algebras over a given field). Section 4 is the bridge between Sections 2 and 3. We show there is a bijection between isomorphism classes of central simple algebras, isomorphism classes of Severi-Brauer varieties, and a Galois cohomology set. The connection between all of the objects is actually really explicit, and we describe various maps going from one class of objects to another. Proving theorems for one class of objects will often lead to theorems for the other class as well. This is useful when one wants to reduce some geometric phenomena to purely algebraic techniques, or vice-versa. At varying levels, and with varying purposes, the material of Sections 2-4 can all be found in [GS06, Mil80, KMRT98, Jah00, Bru, Kol16]. The sources [Jah00, Bru] both approach the subject through the Grothendieck-Weil theory of descent. The source most similar in approach (but far more detailed than appearing here) is [GS06]. An approach that parallels the one taken in these notes, but using schemes instead of varieties, can be found in [Mil80]. An approach that doesn’t limit to twisted forms of projective space, and includes other projective homogeneous varieties, can be found in [KMRT98]. A geometric approach can be found in [Kol16]. 1. Recollections The purpose of this section is to recall concepts and definitions from algebraic geometry and Galois cohomology. A variety will be a separated scheme of finite type over a field k. However, the only varieties we’ll actually consider are projective spaces and subvarieties of these. Now every scheme X determines a functor h X : CRing → Set from the category of commutative unital rings with ring morphisms to the category of sets with set maps; this is just the Yoneda functor composed with the Spec(-) functor so it’s defined on objects by h X (R) = Hom(Spec(R),X) and on morphisms f : R → S by h X (f ) : Hom(Spec(R),X) → Hom(Spec(S),X). 1
Transcript
TWISTED FORMS IN GEOMETRY AND ALGEBRA
EOIN MACKALL
We study objects defined over a base field k, and see how these objects simplify as we move over extensionsof k. This is akin to how a polynomial in one variable may be irreducible over a given field but, over certainfield extensions the polynomial splits into smaller pieces, eventually to linear terms (and over an algebraicclosure, every polynomial completely splits). In the language below, we’d call an irreducible polynomial atwisted form of a product of linear terms.
After recalling some notions from algebraic geometry and cohomology in Section 1, we’ll move to thestudy of Severi-Brauer varieties in Section 2. Severi-Brauer varieties are twisted forms of projective space.This means Severi-Brauer varieties are varieties that become isomorphic to projective space after moving toa finite extension of the base field. As such, they have interesting rationality properties and serve as usefulnontrivial examples in algebraic geometry.
In Section 3 we carry out a study, similar to that of Section 2 but, with the objects of interest beingcentral simple algebras. Central simple algebras are twisted forms of matrix algebras. This means centralsimple algebras are associative (but not necessarily commutative) k-algebras which become isomorphic to amatrix algebra over a finite extension of their base field. Due to Morita theory this takes on the particularlysimple form of the study of division algebras over k (however, it remains a complicated problem to classifydivision algebras over a given field).
Section 4 is the bridge between Sections 2 and 3. We show there is a bijection between isomorphismclasses of central simple algebras, isomorphism classes of Severi-Brauer varieties, and a Galois cohomologyset. The connection between all of the objects is actually really explicit, and we describe various maps goingfrom one class of objects to another. Proving theorems for one class of objects will often lead to theoremsfor the other class as well. This is useful when one wants to reduce some geometric phenomena to purelyalgebraic techniques, or vice-versa.
At varying levels, and with varying purposes, the material of Sections 2-4 can all be found in [GS06,Mil80, KMRT98, Jah00, Bru, Kol16]. The sources [Jah00, Bru] both approach the subject through theGrothendieck-Weil theory of descent. The source most similar in approach (but far more detailed thanappearing here) is [GS06]. An approach that parallels the one taken in these notes, but using schemes insteadof varieties, can be found in [Mil80]. An approach that doesn’t limit to twisted forms of projective space,and includes other projective homogeneous varieties, can be found in [KMRT98]. A geometric approach canbe found in [Kol16].
1. Recollections
The purpose of this section is to recall concepts and definitions from algebraic geometry and Galoiscohomology. A variety will be a separated scheme of finite type over a field k. However, the only varietieswe’ll actually consider are projective spaces and subvarieties of these. Now every scheme X determines afunctor
hX : CRing→ Set
from the category of commutative unital rings with ring morphisms to the category of sets with set maps;this is just the Yoneda functor composed with the Spec(−) functor so it’s defined on objects by
hX(R) = Hom(Spec(R), X)
and on morphisms f : R→ S by
hX(f) : Hom(Spec(R), X)→ Hom(Spec(S), X).1
When X is a variety, our Yoneda functor is defined on a different category than all commutative rings.Instead, a variety X over a field k determines a (similarly defined) functor
hX : Algk → Set
from the category of commutative k-algebras. We’re going to be interested in the restriction of hX to thesmaller category FExtk of field extensions E of k with the choice of an embedding k → E, which we will alsocall hX by abuse of notation.1
Definition 1.1. Let V be a k-vector space. There is a variety called the projective space of V and writtenP(V ). The functor hP(V ) has the following description: for any field extension E/k we have
hP(V )(E) = W ⊂ V ⊗k E : W is a 1-dimensional linear subspace of V ⊗k E
and for any morphism of fields f : E → L we have hP(V )(f) : hP(V )(E)→ hP(V )(L) which takes an elementW to W ⊗E L.
Up to the choice of an isomorphism, the projective space of a vector space depends only on the dimensionof the vector space. If dim(V ) = n+ 1 we write Pn
k for (the isomorphism class of) this variety and call it theprojective space of dimension n over k. We occasionally omit the subscript k when it’s clear from context.
One way to think about this is that projective n-space should classify lines not only over the base field kbut, over all field extensions of k as well. We can similarly describe closed subvarieties of projective space.
Definition 1.2. A projective variety is a closed subvariety of projective space. Given a projective variety Xwith an embedding X ⊂ Pn
k = P(V ), one can realize X as the variety of a finite collection of homogeneouspolynomials f1, . . . , fm in variables x0, . . . , xn.
The functor hX has the following description, relative to hP(V ): for any field extension E/k we have
and for a morphism f : E → L the map hX(f) is the restriction of hP(V )(E) to hX(E).
Lastly, we can change the base a variety is over as follows:
Definition 1.3. Suppose k is a field, X is a variety over k, and E is a field extension of k. There is a varietyXE (determined up to a k-automorphism of E) over E whose functor has the following description: for anyfinite extension L of E (and choice of embedding E → L)
hXE (L) = hX(L)
and for any morphism of fields f : L→ F we have hXE (f) = hX(f).
Since we’re focusing heavily on fields and their extensions it shouldn’t be surprising we will be usingGalois theory often. Recall a finite extension of fields E/k is Galois if E is a normal, separable extension ofthe field k. If E/k is Galois, then we have the fundamental theorem of Galois theory:
Theorem 1.4. Let G = Gal(E/k) = AutFExtk(E). Then there is an inclusion-reversing bijection betweensubgroups H of G and intermediate field extensions F of E/k given by
H 7→ EH and F 7→ Gal(E/F ).
A subgroup H is normal if, and only if, EH is a Galois extension of k; when this happens there is a canonicalisomorphism
Gal(EH/k) ∼= G/H.
1This means every variety is going to determine a functor hX : FExtk → Set defined by hX(E) = Hom(Spec(E), X). Theseare often called the E-valued points of X. However, it should be noted that although we had a Yoneda functor when the domaincategory was Algk this is no longer true when restricted to the category of field extensions of k. For example, X = Spec(k)
and X′ = Spec(k[x]/(x2)) have identical functors hX = hX′ : FExtk → Set. All statements are phrased so that they hold truewhen one considers honest varieties. The functor from fields to sets is meant to act as a device so that one can appreciate some
results obtainable using schemes without needing to understand what a scheme is.
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In what follows we’ll often be in the situation: there is a base field k, a field extension E of k, and acategory of algebraic objects defined over E, AE ; we’ll also have a category of objects Ak over our basefield k; there will be a functor f : Ak → AE which can reasonably be called “base change” or “extension ofscalars”. Galois theory allows us to ask to what extent this functor can be split. More precisely, given anobject S in obj(AE), can we associate an object T in obj(Ak) with f(T ) = S? This doesn’t always work, nordoes it always make sense, but when both of these conditions hold we say that Galois descent holds in thissetting. This terminology is because the object T can often be found using Galois theory. That is to say, Scomes equipped with an action of the Galois group G = Gal(E/k), and T will more than likely be SG, thesubobject of S invariant under the action of G.
This naturally leads one to study, given an arbitrary group G, the functor of invariants, (−)G. Onecan check the functor (−)G is left-exact (on Z[G]-modules say) and, by general homological machinary, onecan study the derived-functors of (−)G. This (hopefully) serves as motivation for why one might studygroup cohomology – introduced below – although the real motivation for the occurance of group cohomologythroughout history is due to its applications in studying Galois modules or, more generally, modules overan arbitrary groupring. Unfortunately, the objects a group can act on are largely variable. This leads tomultiple definitions of group cohomology, which all agree whenever they should (i.e. the group cohomologyof a G-module is the group cohomology of the same object considered as a G-group and this is the same asgroup cohomology of the same object considered as G-set, whenever these are defined).
Definition 1.5. Let G be a group. A G-set is a set A with an action of G.We define the zeroth group cohomology of G with values in a G-set A to be the set
H0(G,A) = AG = x ∈ A : σx = x for all σ ∈ G.
Definition 1.6. Let A be a G-set, and suppose further A is a group. If the G-action commutes with thegroup operation of A (in the sense σ(x ∗ y) = σx ∗ σy for all σ in G and x, y in A), then we say A is aG-group.
We write Z(G,A) for the set of 1-cocycles of G with values in A. By definition, elements of this set areset maps f : G → A which satisfy the condition f(στ) = f(σ) ∗ σf(τ) for all σ, τ in G; the set Z(G,A) ispointed with distinguished element the map eA : G → A sending all σ in G to the identity e of A. We putan equivalence relation on Z(G,A): two 1-cocycles f, g are equivalent, f ∼c g and said to be cohomologous,if there is an element x in A such that f(σ) = x ∗ g(σ) ∗ σx−1 for all σ in G.
We define the zeroth group cohomology of G with values in the G-group A to be the set
H0(G,A) = AG
and the first group cohomology of G with values in the G-group A to be the set of equivalence classes of1-cocycles under the relation of being cohomologous
H1(G,A) = Z(G,A)/∼c.
Definition 1.7. Let A be a G-group, and suppose further A is abelian. Then A is called a G-module. Notesuch A are exactly modules over the groupring Z[G].
We define the ith group cohomology of G with values in the G-module A to be the group
Hi(G,A) = Ri(A)G = RiHomZ[G]−mod(Z, A) = ExtiZ[G](Z, A).
In other words, the ith group cohomology is defined to be the ith right derived functor of the invariantsfunctor (−)G. Equivalently, this is the right derived functor of Hom(Z,−) in the category of modules overthe groupring Z[G] (here the action of G on Z is trivial).
In general, there will not be a long exact sequence of group cohomology attached to a short exact sequenceof G-groups. Of course, this will happen if all of the objects in the short exact sequence are G-modules, butthis won’t be the case in general. Instead, we have the following result:
Proposition 1.8. If B is a group and A a subgroup of B both with compatible G-actions, then there is ashort excact sequence of pointed G-sets
1→ A→ B → B/A→ 13
to which one can associate an exact sequence of pointed sets
1→ H0(G,A)→ H0(G,B)→ H0(G,B/A)→ H1(G,A)→ H1(G,B).
If A is normal in B, then the short exact sequence above is a short exact sequence of G-groups. In thissetting, the associated exact sequence of pointed sets extends
If A is central in B, then the short exact sequence above is a short exact sequence of G-groups. In thissetting, the associated exact sequence of pointed sets reads
Remark 1.9. Recall if (A, a), (B, b), (C, c) are pointed sets then a sequence
Af−→ B
g−→ C
with f(a) = b, g(b) = c is said to be exact at B if f(A) = g−1(c). A sequence is said to be exact if it is exactat all intermediate steps.
Consequently, if
Af−→ B → 1
is an exact sequence of pointed-sets then f is surjective. However, if
1→ Af−→ B
is an exact sequence of pointed-sets then f need not be injective.
Although all of the above applies to an arbitrary group G, we will work with a more specific type ofgroup – a profinite group – which will have more information attached to it that needs to be accountedfor. Our primary example of a profinite group will be the Galois group of a Galois field extension, and theextra information attached to this group can be seen in the fundamental theorem of Galois theory. All finitegroups will be profinite groups in an obvious way. The differences only become apparent for infinite Galoisextensions so, we will need to recall the corresponding fundamental theorem of infinite Galois theory.
Theorem 1.10. Let k be a field, Ω a Galois extension of k (finite or infinite), and G = Gal(Ω/k) theGalois group of Ω/k. The group G is a profinite group (this means it is the inverse limit, in the category oftopological groups, of its finite quotient groups equipped with the discrete topology).
There is an inclusion-reversing bijection between the set of closed subgroups H of G and intermediatefields F of Ω/k given by
H 7→ ΩH and F 7→ Gal(Ω/F ).
A closed subgroup H of G is open if and only if ΩH is a finite extension of k. A closed subgroup H of G isnormal if, and only if, ΩH is Galois over k. If H is normal, there is a canonical isomorphism
Gal(ΩH/k) ∼= G/H.
This somehow suggests if we want to study the cohomology of infinite Galois groups, while retaininginformation on the field extensions of our base field, then we should take into account which subgroups areclosed as well.
Definition 1.11. Let G be a profinite group. Let A be a discrete topological space. We say A has acontinuous G-action if the stabilizer of any element of A is an open subgroup of G. We call A a continuousG-set when it has a continuous G-action.
We define the zeroth profinite cohomology of G with values in a continuous G-set A to be the set
H0pro(G,A) = AG.
Definition 1.12. Let A be a G-set, and suppose further A is a group. If the action of G commutes withthe group operation of A then we call A a continuous G-group.
Similar to before, we write Zpro(G,A) for the set of all continuous 1-cocycles of G with values in A. Here acontinuous 1-cocycle of G is a continuous map f : G→ A which satisfies the condition f(στ) = f(σ) ∗σf(τ)for all σ, τ in G; the set Zpro(G,A) is pointed with distinguished element the map eA : G → A sending all
4
σ in G to the identity e of A. We put an equivalence relation on Zpro(G,A) the same way as before: two1-cocycles f, g are defined to be equivalent, f ∼c g and said to be cohomologous, if there is an element x inA such that f(σ) = x ∗ g(σ) ∗ σx−1 for all σ in G.
We define the zeroth profinite cohomology of G with values in the continuous G-group A to be the set
H0pro(G,A) = AG
and the first profinite cohomology of G with values in the continuous G-group A to be the set
H1pro(G,A) = Zpro(G,A)/ ∼c .
Definition 1.13. Let A be a G-group, and suppose further A is abelian. Then A is called a continuousG-module.
We define the ith profinite cohomology of G with values in a continuous G-module A to be the colimit
Hipro(G,A) = lim−→
N
Hi(G/N,AN )
where N runs over the open normal subgroups of G and the colimit is taken over the canonical (“inflation”)maps Hi(G/N,AN )→ Hi(G,A) of group cohomology.
Again we need to adjust our definition of exact sequence and the corresponding long exact sequence inGalois cohomology.
Proposition 1.14. If A ⊂ B is an inclusion of continuous G-groups, then there is a short excact sequenceof pointed continuous G-sets
1→ A→ B → B/A→ 1
to which one can associate an exact sequence of pointed sets
1→ H0pro(G,A)→ H0
pro(G,B)→ H0pro(G,B/A)→ H1
pro(G,A)→ H1pro(G,B).
If A is normal in B, then the short exact sequence above is a short exact sequence of continuous G-groups.In this setting, the associated exact sequence of pointed sets extends
1→ H0pro(G,A)→ H0
pro(G,B)→ H0pro(G,B/A)→ H1
pro(G,A)→ H1pro(G,B)→ H1
pro(G,B/A).
If A is central in B, then the short exact sequence above is a short exact sequence of continuous G-groups.In this setting, the associated exact sequence of pointed sets extends
1→ H0pro(G,A)→ H0
pro(G,B)→ H0pro(G,B/A)→ · · ·
· · · →H1pro(G,A)→ H1
pro(G,B)→ H1pro(G,B/A)→ H2
pro(G,A).
This should be all of what we need for this survey. However, if you’re interested in learning more about theabove: for more background on varieties one can look at [Kun13, Chapter 1] and [Mil17a], or [FGI+05, Vis08,Chapters 1-2] for a categorical point of view; for information on extending Galois theory to infinite extensionsone can look at [Mil17b, Chapter 7]; for more detail on the construction of Galois cohomology one can lookat [Ser97], [Mil13a, Chapter 2], [KMRT98, Chapter VII], or [GS06, Chapters 3-4].
Notation. Given a variety X over a field k we write X for both the variety and the functor hX . So, if E is afield extension of k then byX(E) we really mean the set hX(E). For example, Pn(E) = P(V )(E) = hP(V )(E)where V is a n+ 1 dimensional vector space over k.
From now on we fix a field k and a separable closure ksep of k. We write Gk for the absolute Galoisgroup Gal(ksep/k). We also omit the subscript pro when writing profinite cohomology. If we are workingwith the profinite cohomology of Gk with values in A, we call it Galois cohomology and write Hi(k,A) forthe corresponding Galois cohomology sets. If we want to denote a pointed element without name, we willwrite ∗ for this element.
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2. Twisted Forms of Projective Space
In this section we work through essentially only one concrete example of a Severi-Brauer variety: a conicin P2
R. We develop very select pieces of the general theory, and show how they can be seen in our exampleof a conic.
Definition 2.1. A Severi-Brauer k-variety is a k-variety X such that, over a finite field extension E ofk, there is an isomorphism XE
∼= PdE for some nonnegative integer d. The choice of an isomorphism
f : XE → PdE is called a splitting of X.
Remark 2.2. Note, one can replace the requirement X is a k-variety by the requirement X is an arbitraryk-scheme. Actually, if X is a k-scheme such that XE
∼= PdE , then X is necessarily a k-variety.
As stated, an equivalent formulation of the definition of a Severi-Brauer variety is: a variety that becomesisomorphic to projective space over an algebraic closure of the base field. Although one direction is clear(if a Severi-Brauer variety is isomorphic to projective space over a finite extension then it is isomorphic toprojective space over an algebraic closure), the converse is not trivial. This relies on the fact Severi-Brauervarieties are projective (this follows from Galois descent [Jah00, Lemma 2.12]; this also follows from knowingthe existence of various projective embeddings; see Corollary 2.8 or the discussion above Theorem 4.8 below).
Using that Severi-Brauer varieties are projective, we can prove the converse like so: let X be a Severi-Brauer k-variety and X ∼= Proj(k[x0, . . . , xn]/(f1, . . . , fm)). An isomorphism Xk → Pd
k= Proj(k[T0, . . . , Td])
then sends the d+ 1-coordinate functions T0, . . . , Td to polynomials p0, . . . , pd in k[x0, . . . , xn]/(f1, . . . , fm).But there are only finitely many coefficients αi in the polynomials p0, . . . , pd so this isomorphism is alsodefined over the field k(αi) which is a finite extension of k.
In fact, it suffices to only consider finite Galois extensions in Definition 2.1. Equivalently, a Severi-Brauervariety is a variety that becomes isomorphic to projective space over a separable closure of the base field.This is harder to see than the above. It follows from Theorem 4.8 below and the same statement for centralsimple k-algebras, [Jac96, Theorem 1.6.19]. Alternatively, one can first prove Theorem 4.9 below by differentmethods, [GS06, Theorem 5.1.3], then observe any geomtrically reduced k-variety X has a point over ksep. Tosee this last claim, note that it is more generally true the subset of points whose residue field is a separableextension of k is a dense open subset of X. This implies Xksep contains a dense open subset of rational
points.2 To prove this, one uses that X being geometrically reduced implies the residue field at any genericpoint of X is separable over k, [Sta17, Tag 030W]. Hence by [Sta17, Tag 056V] the smooth locus of X isa dense open subset and by [Sta17, Tag 00TV] any point x in this locus has residue field k(x) that is aseparable extension of k.
Example 2.3. Projective space Pnk is always a Severi-Brauer variety over k. This is often called the trivial,
or split, case.3
The next example is our first example of a nontrivial Severi-Brauer variety.
2This is a useful observation about rational points of varieties over extensions of the base field that isn’t explicitly said
anywhere. Let X be a variety over k and E a finite extension of k with X(E) not empty. This implies there is a point x ∈ Xwhose residue field k(x) has an embedding into E. Let U = Spec(A) be an affine open containing x and let m be the maximal
ideal corresponding to x. Then, by the construction of fibered products, A⊗k E is an open subset of XE . There is a surjectionA⊗k E → k(x)⊗k E since E is a flat k-module and composing this with the k-linear multiplication map k(x)⊗k E → E showsthere is a maximal ideal n of A⊗k E whose residue field is E. Thus, XE has an E-rational point.
More generally, a similar approach allows one to characterize fibered products of schemes. Let X,Y, Z be three schemes withmaps f : X → Z and g : Y → Z. Then X ×Z Y can be described as the set of quadruples (x, y, z, p) where f(x) = g(y) = z
and p is a prime ideal of k(x)⊗k(z) k(y), [Sta17, Tag 01JT].3Kind of funnily, a nonexample of a Severi-Brauer variety over k is Pn
E whenever E is a finite nontrivial extension of k.
If E is separable then this is because PnE is not geometrically connected as a k-variety but, every Severi-Brauer variety X is
geometrically connected since Xk∼= Pn
kfor some n. To see Pn
E is not geometrically connected, note PnE = Proj(E[x0, . . . , xn])
and Spec(k) = Proj(k[y]). Then (PnE)k = Proj(E ⊗k k[x0 ⊗ y, . . . , xn ⊗ y]) and, by the Primitive Element Theorem, E ⊗k k =
k[x]/f ⊗k k = k[x]/f =∏deg f
i=1 k so that (PnE)k =
∐deg fi=1 Pn
k. Here we used the formula: for any graded k-algebras R = ⊕nRn,
S = ⊕nSn we have Proj(R)×k Proj(S) = Proj(⊕nRn ⊗k Sn).
If E is inseparable, then we can write E/F/k where F/k is a separable extension and E/F is a nontrivial purely inseparableextension. Moving to a separable closure, we can assume F = k = ksep and to see Pn
Example 2.4. Let X be the subvariety of P2R defined by the vanishing of the polynomial x2 +y2 +z2. Since
X is one dimensional, and R has only one nontrivial finite extension, to see X is a nontrivial Severi-Brauervariety we just need to check X 6∼= P1
R and XC∼= P1
C.The first of these two requirements is easy to see since X has no R-rational points (we’re in R so
x2 + y2 + z2 = 0 if and only if x = y = z = 0 which isn’t considered when constructing projective space). Soit’s clear X 6∼= P1
R.The second is more difficult. To see XC
∼= P1C we will find an isomorphism between the two. We’ll do
this by using the Riemann-Roch theorem and some facts about curves. For the rest of this example allcohomology is coherent sheaf cohomology.
First, observe XC is a smooth curve. To check this we can use the Jacobian criterion [Liu02, Chapter 4,Theorem 2.19] to check XC is regular at all of its closed points (which is an equivalent condition for XC).How we do this: look at XC in the affine open subsets of P2
C given by the nonvanishing of x, y, or z; next,check that the Jacobian matrix of XC is everywhere rank 1 in these opens (e.g. when z is nonvanishing theJacobian is (
2x/z 2y/z 0)
and this is rank 0 only when x = y = 0 which never happens on the curve (x/z)2 + (y/z)2 + 1 = 0. Thesame argument works in the other cases). Now observe XC is genus 0 by the degree-genus formula since0 = (2− 1)(2− 2)/2.4 Finally, note there is a rational point p on XC (e.g. p = (1, 0, i) works).
This puts us in position to prove the statement. We’ll use the Riemann-Roch formula for the divsor p,
dimH0(XC,O(p))− dimH0(XC, ωXC⊗O(−p)) = deg(p)− g + 1.
Note deg(p) = 1, g = 0 as we saw above, and H0(XC, ωXC⊗ O(−p)) = 0; this last equality is because, by
definition, this is (up to isomorphism) the vector space of rational functions f such that div(f)+div(s)−p ≥ 0where div(s) is the difference of the divisor of some nonzero global section of ωXC
⊗ OXC(n) and nH for
some hyperplane section H (since the divisor of any rational function f on a projective curve has degree 0[Har77, Corollary 6.10], deg(ωXC ) = 2g − 2 = −2 by Riemann-Roch, and deg(p) = 1 this sum will never be“positive”). All together this shows
dimH0(XC,O(p)) = 1− 0 + 1 = 2
and consequently there is a nonconstant rational function on XC, defining a map f : XC → P1C.5
What remains is to show f is the desired isomorphism. Since f is nonconstant it follows that ordp(f) = −1(if ordp(f) = 0, which is the only other possible case, then we’d have ordz(f) = 0 for all closed points z ∈ XC
which implies f is a constant). We can use [Har77, Proposition 6.9], which says
deg(f∗z) = deg(f) deg(z)
4There are two definitions of genus for a variety X: the arithmetic genus pa(X) = (−1)dim(X)(χ(X,OX)−1) and geometric
genus pg(X) = dimH0(X,ΩX/k). When X is a smooth projective curve, they are equal (and when X is a complex curve they
are equal to the topological notion of genus). If g is the genus of a smooth projective plane curve X ⊂ P2, and d is the degreeof X, the degree-genus formula says g = (d− 1)(d− 2)/2. It follows from the adjunction formula: if i : D → V is the inclusion
of a smooth hypersurface into a smooth variety then there is an isomorphism ωD∼= i∗(ωV ⊗OV (D)).
Here’s a proof of the degree-genus formula for the inclusion i : X → P2 of a smooth curve. By the Riemann-Roch formulafor locally free sheaves of rank 1 we have equality
dimH0(X,ωX)− dimH0(X,ω∨X ⊗ ωX) = deg(ωX) + 1− g.
Note that dimH0(X,ωX) = pg(X) = g and dimH0(X,ω∨X ⊗ ωX) = dimH0(X,OX) = 1. To prove the claim, we’ll compute
deg(ωX) using the adjunction formula. We choose an isomorphism CH(P2) = Z[h]/(h3) where h is the class of a hyperplane
in P2. Then deg(ωX) is the coefficient of h2 in the pushforward i∗c1(ωX) ∈ CH0(P2). By the functorality of chern classes,i∗c1(ωX) = i∗c1(i∗(ωP2 ⊗ OP2 (X))) = i∗(i∗c1(ωP2 ) + i∗c1(OP2 (X))) = (−3d + d2)h2. This last equality isn’t necessarily
trivial, and it has to be checked essentially by hand. All together we find,
g − 1 = (d2 − 3d) + 1− g
from which the formula follows by solving for g.5There are two ways of defining the map f here. One can either use that the divsor p gives a complete linear system of
projective dimension 1 or, one can use that a nonconstant rational function h on XC defines a rational map to P1 through the
composition C(XC) = lim−→U⊂XCHom(U,A1
C) ⊂ lim−→U⊂XCHom(U,P1
C) where the colimit runs through open subvarieties U .
Technically, this only defines a map g from a dense open subset of XC. However, g can be uniquely extended to a map f
defined on all of XC by the algebraic Hartog’s Lemma [GW10, Theorem 6.45].
7
for any closed point z of XC to show the degree of f is 1. We just pick a specific point, like z = ∞, wherewe know f∗z = p (this last equality is from the definition: we pick a local parameter like 1/t at ∞ andcompute ordp(f(1/t)) = −ordp(f(t)) = −(−1) = 1). Then this shows deg(f) = 1 which, by definition,means [C(P1
C) : C(XC)] = 1. Equivalently, the map induced by f on function fields is an isomorphism. Bythe equivalence between function fields (of curves) and curves, [Sta17, Tag 0BY1], this shows f is in fact anisomorphism.
A careful reading through the last example actually shows we’ve found the following corollary.
Corollary 2.5. Let X be a smooth projective curve of genus 0 over a field k. If X has a rational point, thenX is isomorphic with P1
k.
Remark 2.6. [Art82, Section 2, (2.1)]. It’s possible to compute the Picard group of any Severi-Brauervariety using the Hochschild-Serre spectral sequence [Mil13b, Theorem 14.9]. This is a spectral sequencefrom group to etale cohomology
Er,s2 = Hr(G,Hset(Y,F|Y )) =⇒ Hr+s
et (X,F)
where Y → X is a Galois covering with Galois group G, and F is a sheaf on the etale site of X. In oursituation, we set X to be a Severi-Brauer variety over k, E a Galois extension splitting X, and we assume Xhas positive dimension (the Picard group of a zero-dimensional Severi-Brauer variety is trivial, see Remark2.7; alternatively, one can prove this directly since a zero-dimensional Severi-Brauer variety is the spectrumof a local ring, hence any locally free sheaf of rank 1 is actually free). We set Y = XE and let the mapY → X be the projection. Finally, we let F = Gm.
The five-term exact sequence associated to the above spectral sequence reads
0→ E1,02 → Pic(X)→ E0,1
2 → E2,02 → H2(X,Gm)
so if we know E1,02 , E0,1
2 , and E2,02 we’re done. Essentially each of these has been computed classically.
E1,02 = H1(G,H0
et(Y,Gm)) = H1(G,E×) = 0
where all of the equalities are from definition except the last, which follows from Hilbert’s Theorem 90 inGroup cohomology, see Lemma 4.10 below.
E0,12 = H0(G,H1
et(Y,Gm)) = H0(G,Pic(Y )) = Pic(Y )G = Pic(Y )
where the second equality follows from the isomorphisms H1et(Y,Gm) ∼= H1
Zar(Y,Gm) = Pic(Y ) and the lastequality can be checked (i.e. since Y ∼= Pn and Pic(Pn) = Z · [O(−1)] we can let f be an automorphism ofPn and observe f∗O(−1) has no global sections. Then the action of f on the Picard group must take theisomorphism class [O(−1)] to a class generating the group and without global sections, which can only beitself).
E2,02 = H2(G,H0
et(Y,Gm)) = H2(G,E×) = Br(E/k)
where the only equality to explain is the last one, which will be discussed later in Remark 4.6.All together this implies there is an exact sequence
0→ Pic(X)→ Pic(Y )δ−→ Br(E/k).
This identifies Pic(X) ⊂ Z · [O(−1)] ∼= Pic(Y ) as a free abelian group of rank 1; this map is even the naturalpullback map on sheaves. The image of δ is a finite cyclic group, and its size is often called the exponent ofthe Severi-Brauer variety X.
Remark 2.7. So far, we can already say some things about the Chow ring of a Severi-Brauer variety.Let X be one such variety. If X is dimension 0, then CH(X) = Z. To see this, we can just classify all0-dimensional Severi-Brauer varieties. By Galois descent, Lemma 4.3 below, X must be affine. X must thenbe the spectrum of an Artinian ring since X has a finite underlying set. In other words, X = Spec(R) whereR ∼= k⊕n as k-modules for some n ≥ 1 but, of course n = 1 since it is after moving to an algebraic closure.Thus, the only 0-dimensional Severi-Brauer variety is trivial.
Since Severi-Brauer varieties are smooth, Remark 2.6 completely computes the Chow ring for any Severi-Brauer variety of dimension 1 (for a smooth variety X there is a canonical isomorphism Pic(X) ∼= CH1(X)
given by the first Chern class). For example, Remark 2.9 below shows the conic X in example 2.4 hasCH(X) = CH0(X)⊕ CH1(X) = Z⊕ 2Z.
For Severi-Brauer varieties of dimension 2, one can again completely compute the Chow ring. If X is aSeveri-Brauer variety of dimension 2, then CH(X) = CH0(X) ⊕ CH1(X) ⊕ CH2(X) where CH0(X) = Z,CH1(X) = eZ, and CH2(X) = dZ where e > 0 is the exponent of X defined as in Remark 2.6. The numberd > 0 is called the index of X: it is the gcd of the degrees of all closed points on X. This follows from[CM06, Proposition 4.4].
In higher dimension, the Chow ring of a Severi-Brauer variety can contain torsion (see [Kar98] for firstresults in this direction).
As a corollary of Remark 2.6, we get that Severi-Brauer varieties are projective.
Corollary 2.8. Let X be a Severi-Brauer variety over k of dimension n > 1. Let d be the exponent of X,as defined in Remark 2.6. There exists a closed embedding
ρ : X → PN−1k
where N =(n+dd
)such that ρksep is the degree d Veronese embedding.
Proof. Let E be a finite Galois extension splitting X. Write
π : XE → X and f : PnE∼−→ XE
for the projection and for a chosen splitting of X respectively. Remark 2.6 shows there exists a locally freesheaf L of rank 1 on X such that (f π)∗L ∼= OPn(d). We’ll use L to construct the embedding ρ.
By flat base change [Sta17, Tag 02KH], there’s an isomorphism of coherent sheaf cohomology
Hi(X,L)⊗k E ∼= Hi(XE , π∗L)
for all i ≥ 0. In particular, this holds true for i = 0. Pulling back by f gives another isomorphism, hence wefind
H0(X,L)⊗k E ' H0(XE , π∗L) ∼= H0(Pn
E ,OPn(d)).
Since H0(X,L) is an k-vector space, and H0(PnE ,OPn(d)) is a finite dimensional E-vector space, it follows
H0(X,L) has a basis consisting of a finite number of global sections s0, . . . , sr that pullback to the basisof monomials in H0(Pn
E ,OPn(d)). Moreover, s0, . . . , sr globally generate L. To see this, note π f issurjective (base change of a surjection is a surjection). So, if y = π f(x) is a point in X then: s0, . . . , srsimultaneously vanishing at y implies, by the isomorphism (π f)∗Lx ∼= Lπf(x) ⊗OX,πf(x) OPn,x, that the
pullbacks (π f)∗s0, . . . , (π f)∗sr simultaneously vanish at x (and this never happens because we chose the(π f)∗si so that it wouldn’t).
We can then construct ρ : X → PN−1k to be the map corresponding to the morphism defined by the si,
[Har77, Chapter 2, Theorem 7.1]. After base change to E, ρE is isomorphic to the Veronese embedding
PnE → PN−1
E by construction. To complete the proof, it remains to check ρ is a closed immersion. This canbe checked locally on the target, and after making a faithfully flat extension of the base field, hence we canconclude since ρE is a closed immersion.
Remark 2.9. The classification of zero-dimensional Severi-Brauer k-varieties turned out to be trivial: thereis only the point Spec(k). In dimension 1, that is for Severi-Brauer curves over k, we can again obtain acomplete classification in an appropriate sense.
To do this, we first need to observe the exponent of a Severi-Brauer curve X over k is less than or equalto 2. Let E be a field extension splitting X, π : XE → X the projection, and let f : P1
E → XE be a choiceof splitting of X. The cotangent sheaf Ω1
X/k is locally free of rank 1 since X is smooth of dimension 1. It
also pulls-back nicely under base change [Sta17, Tag 01V0], π∗Ω1X/k ' Ω1
XE/E, and there is an isomorphism
Ω1P1E/E
∼= OP1(−2). Dualizing gives us a locally free rank 1 sheaf Ω∨X/k that pulls back to OP1(2) under
π f . There may or may not be another sheaf with pullback OP1(1), so at best we can say the exponent is2 or less.
Assuming X is not trivial, corollary 2.8 provides us with an embedding of our Severi-Brauer curve X intoP2k. Since the genus of X is preserved under field extension, X must have genus 0. The genus-degree formula
then says X has degree 2 (since we are assuming X 6∼= P1). Moreover, if k does not have characteristic 2,9
then X can be realized as the vanishing set of a polynomial ax2 + by2 = z2 under an appropriate choice ofcoordinates x, y, z on P2
k (because any quadratic form in characteristic not 2 can be diagonalized). Note,since we had to choose coordinates, the pair (a, b) is not uniquely determined by X; there could be a differentchoice of coordinates giving a different pair (c, d). Conversely, every polynomial ax2 + by2 = z2 with a, b ink× determines a Severi-Brauer curve in P2
k (this can be shown following essentially the same argument asExample 2.4).
The space of all degree 2 curves (up to scaling) in P2k is in bijection with the rational points of a 5-
dimensional projective space P5(k). The description above determines the subset of curves that are Severi-Brauer varieties; interestingly, it has the structure of rational points of a locally closed subvariety.
From the definition, it’s clear Severi-Brauer varieties are phenomena that appear only when consideringarbitrary base fields (over an algebraic closure all Severi-Brauer varieties are projective space). One can askwhat general condition an algebraically closed field has that forces a Severi-Brauer variety to be trivial. Inother words, we want to single out some property P of varieties over an algebraically closed field so that: ifan arbitrary Severi-Brauer variety X satisfied P , then X is trivial. The property P we’re searching for isactually the existence of a rational point (by Hilbert’s Nullstellensatz every variety X over an algebraicallyclosed field has a rational point).
Theorem 2.10. A Severi-Brauer variety X over k is trivial if, and only if, it has a k-rational point.
Proof. One direction is clear, since if X ∼= Pnk then X has a k-rational point.
The converse takes more work to show, and won’t be completed until Theorem 4.9 below.
3. Central Simple Algebras
In this section we again work through essentially only one concrete example but, this time our objects arecentral simple algebras. The example of this section is a quaternion R-algebra; this is closely related to theconic in Example 2.4.
Definition 3.1. A central simple k-algebra is an associative (but not necessarily commutative) k-algebraA such that, over a finite field extension E of k and for some r, there is an isomorphism A ⊗k E ∼= Mr(E)with the E-algebra of size r × r matrices.
Remark 3.2. We can make some immediate observations. Any central simple k-algebra has square k-dimension; this is because dimension is preserved under extension of the base field. The positive squareroot of the k-dimension of A is called the degree of A and written deg(A). Also, central simple algebras arepreserved under field extension: if A is a central simple k-algebra, and E/k is a finite field extension, thenA⊗k E is a central simple E-algebra.
Remark 3.3. Since Mr(E) ⊗E k ∼= Mr(k) we see that it is necessary a central simple algebra splits overan algebraic closure of the base field. Since algebraic closures are colimits of finite extensions, this is also asufficient condition (i.e. if there is an isomorphism A ⊗k k ∼= Mr(k) then there are elements xij ∈ A ⊗k kmapping to the standard matrices eij with 1 in the i, jth position and 0’s elsewhere. Since only finitely
many elements al of k are required to define the elements xij , this also defines an isomorphism over the finiteextension containing all of the al). It’s a nontrivial fact that a central simple algebra splits over a finiteseparable extension, [Jac96, Theorem 1.6.19], and hence also over a finite Galois extension. By a similarargument to the above, it is equivalent that A splits over a separable closure of k.
Example 3.4. The matrix algebra Mr(k) is always a central simple algebra over k. This is the trivial, orsplit, case.
A central simple k-algebra A which becomes trivial over a finite extension E/k is said to be split over thefield E. We also use the terminology “A splits over E” if over E there is an integer r and an isomorphismA⊗k E ∼= Mr(E). We call the choice of such an isomorphism a splitting of A.
Example 3.5. Let Q be the real quaternion algebra; this is the R-vector space with basis 1, σ, τ, στ andmultiplication relations σ2 = τ2 = −1, στ = −τσ (extended linearly). To show Q is a nontrivial centralsimple R-algebra it suffices to check only that Q 6∼= M2(R) and Q⊗R C ∼= M2(C).
10
The first of these two requirements is somewhat easy to see, because all of the elements of Q haveinverses while M2(R) has zero divisors. That every element of Q is invertible follows from the computation(a+ bσ + cτ + dστ)(a− bσ − cτ − dστ) = a2 + b2 + c2 + d2 where a, b, c, d are in R.
To see Q ⊗R C is isomorphic with M2(C) we can define an explicit isomorphism. Let eij be the matrixwith 1 in the i, jth position and 0 elsewhere. Define the map f : Q ⊗R C → M2(C) by f(1) = e11 + e22,f(σ) = e12 − e21, f(τ) = ie12 + ie21 and extended C-linearly. A computation shows f is, by construction,a k-algebra homomorphism. It’s also not too difficult to check f is a bijection (since, for instance, it issurjective as a map between 4-dimensional vector spaces), hence an isomorphism.
It may currently be a mystery to the reader as to why such algebras are called central simple. We’ve gonein reverse chronological order defining an object not by its original definition but by a theorem only foundlater. The next proposition shows the name is appropriate.
Proposition 3.6. Let A be an associative k-algebra. Then the following statements are equivalent:
(1) A is central over k (i.e. if xy = yx for all y in A, then x is in k), and simple as an algebra (i.e. theonly two-sided ideals of A are (0) and A)
(2) the map A⊗k Aop → Endk(A) taking a⊗ bop to the linear map f(x) = axb is an isomorphism(3) over an algebraic closure k there is an isomorphism A⊗k k ∼= Mr(k) for some r(4) there is a finite dimensional division algebra D, central over k, and an integer r with A ∼= Mr(D).
Moreover, there is an isomorphism D ∼= EndA(M) for any simple left A-module M , so the isomor-phism class of D is uniquely determined.
Reference. [Mil13a, Chapter 4].
Remark 3.7. By (1) of Proposition 3.6, a division algebra D (in this note a division algebra over k is analgebra, finite dimensional as a vector space, in which every nonzero element has an inverse) with center kis a central simple k-algebra. However, this isn’t immediately clear if one takes their definition of a centralsimple algebra to be (3) of Proposition 3.6 (as we did).
Remark 3.8. Two central simple algebras are Brauer equivalent if they are both isomorphic to matrixalgebras (of possibly different sizes) over isomorphic division algebras.6 Equivalently, a central simple k-algebra A is Brauer-equivalent to a central simple k-algebra B if there are integers r, s such that A⊗kMr(k) ∼=B ⊗kMs(k). Let Br(E/k) be the set of isomorphism classes of central simple algebras over k that split overa finite extension E (if A is a central simple k-algebra then [A] will denote its class). The tensor product ofalgebras induces a well-defined binary operation on the set Br(E/k).
From the natural isomorphisms of a tensor product, the operation defined in the previous paragraph isassociative. There is a unit object, given by the class [k], and inverses exist by (2) of Proposition 3.6. Thus,the set Br(E/k) is equipped with the structure of a group. We call the group Br(E/k) with this operationthe Brauer group of E-over-k.
Given two fields E,F with F/E/k we can define a functorial pullback Br(E/k) → Br(F/k) given bysending [A]→ [A]. This defines an inductive system running over all finite separable extensions of k and wedefine the Brauer group of k, written Br(k), to be the colimit over this system:
Br(k) := lim−→F⊃k
Finite Galois
Br(F/k)
It’s often difficult to compute Br(k) for any particular choice of field k but, there are some nontrivialresults:
Br(k) = 1 by definition
Br(Fq) = 1 by Wedderburn’s theorem
Br(R) = Z/2Z by a theorem of Frobenius
Br(Qp) = Q/Z by local class field theory, [Mil13a, Chapter 4, §4]
Br(C(t)) = 1 for any C1 field the Brauer group is trivial
6This is also called Morita-equivalence but the two notions are defined differently.
11
We will see, in Section 4, the correspondence between Severi-Brauer varieties and central simple algebras.For now, we prove the analogue of Proposition 2.8 for central simple algebras.
Proposition 3.9. A central simple algebra A over k is split if, and only if, it has a left ideal of dimensiondeg(A).
Proof. The “only if” part is straightforward since, if A = Mr(k), one can take the ideal generated by matriceswith with zeros outside a single fixed column.
Conversely, one can show any finite dimensional module over a central simple algebra is a sum of simplemodules, [Mil13a, Chapter 4, Proposition 1.17]. Moreover, if A ∼= Mr(D) is an isomorphism as in Proposition3.5, then one can say exactly what the simple modules of A are: they are column vectors (isomorphic withDr). These can be identfied as subsets of Mr(D) given by a single column of possibly nonzero entries, withzeros elsewhere. The claim then follows immediately from definitions since, if I is a left ideal (it suffices toassume I is minimal) of dimension deg(A), then
dimk(I) = rdimk(D) = r√
dimk(D) = deg(A).
Dividing by r shows dimk(D) = 1, and therefore D = k.
4. A Cohomological Approach
In Section 2 we (briefly) looked at Severi-Brauer varieties, and in Section 3 we (briefly) looked at centralsimple algebras. In this section, we tie these objects together with cohomology. The benefit from doing sois manifold: this will allow us to use computational tools, such as exact sequences and the Brauer group, tostudy varieties and, conversely, it will allow us to use geometry to study purely algebraically defined objects.Most of the proofs in this section closely follow those in [Jah00].
The basis for all of what follows is the observation:
Proposition 4.1. For any field extension E of k there is an isomorphism
PGLn(E) ∼= AutalgE (Mn(E))
defined by [x] 7→ Int(x)(−) := (y 7→ xyx−1) and an isomorphism
PGLn(E) ∼= AutVarE (Pn−1E )
defined by [x] 7→ x#.
In other words, the objects we’ve been studying (Severi-Brauer varieties and central simple algebras) areboth related by the fact they are twisted forms of objects that have the same automorphism group.
Proof. Recall PGLn(E) is defined as the quotient of GLn(E) by its center, Z(GLn(E)) ∼= Gm(E) = E×.The elements of the center consist of constant diagonal matrices, which gives the identification indicated.Consequently, an element γ of PGLn(E) can always be represented γ = [x] as the equivalence class of anelement x of GLn(E).
We first prove the statement for algebras. Let A = Mn(E). The map [x] 7→ Int(x)(−) is well-definedsince, by choosing a different representative for [x], say y = xc with c central, we get
Int(x)(z) = xzx−1 = xczc−1x−1 = Int(y)(z)
for all z in A.Injectivity of the map [x] 7→ Int(x)(−) is nearly immediate since, if [x], [y] define the same inner automor-
phisms thenxzx−1 = yzy−1
for all z in A (hence the same equality holds for all z in GLn(E), since A contains this group as a propersubset). In particular, y−1x is an element of Z(GLn(E)) implying the equality [x] = [y].
Let λ be an E-algebra automorphism of A. Our goal will be to show λ is in the form Int(x)(−) for somex in A; this shows surjectivity of the map [x] 7→ Int(x)(−). By Proposition 3.6.(2), there is an isomorphismA⊗E Aop ∼= EndE(A) given by sending a⊗ bop to the endomorphism g defined by the rule g(x) = axb for allx in A. There is also an isomorphism A⊗E Aop ∼= EndE(A) given by sending a⊗ bop to the endomorphismg′ defined by the rule g′(x) = λ(a)xb for all x in A. This defines two A ⊗E Aop-module structures on
12
A, which we call (A, g) and (A, g′) respectively. Moreover, there must be an isomorphism (A, g) ∼= (A, g′)as A ⊗E Aop-modules by the characterization of the module category of central simple algebras (this wasdiscussed in the proof of Proposition 3.9). In other words, there is an isomorphism ι : A→ A which carriesthe first action of A⊗E Aop to the second. For any matrix x in A, a clever observation shows
ι(x) = ι(1⊗ x · 1) = 1⊗ x · ι(1) = ι(1)x
where 1 is the identity matrix in A. Since ι is an isomorphism, ι(1) must be invertible, in other words ι(1)is a matrix in GLn(E). But, another clever observation shows
Multiplying on the right by ι(1)−1 yields λ(x) = ι(1)xι(1)−1, which is exactly what we wanted to prove.We next prove the statement for varieties. This is done in [Har77, Example 2.7.1.1] but, we include
the argument here for completeness. Recall that, given a matrix A = (aij)ni,j=1 of GLn(E) there is an
automorphism of the graded ring E[x1, . . . , xn] defined by xi 7→∑j aijxj ; we write A# for the induced map
on varieties. The matrix A and the matrix λA, where λ is in E×, define the same maps on the associatedprojective variety, so there is a well-defined map from PGLn(E) to Aut(Pn−1
E ) given by [x] 7→ x#.Injectivity of [x] 7→ x# is nearly immediate since, if [x], [y] define the same automorphism of Pn−1, then
[y−1x] maps the vector (x1, . . . , xn, x1 + · · ·+xn) to a scalar multiple of itself. But this exactly means [y−1x]is the trivial map, and thus [y] = [x].
To see the surjectivity of [x] 7→ x# we look at the action of a matrix on the sheaf OPn−1(1). Thatis, x#∗OPn−1(1) is a sheaf generating Pic(Pn−1) with global sections, hence x#∗OPn−1(1) = OPn−1(1).Hence x# induces an automorphism γ on the E-vector space of global sections Γ(Pn−1,OPn−1(1)) =spanEx1, . . . , xn. Any such γ corresponds uniquely to an element of GLn(E) and, in the chosen basis γ = x.If λ is now an arbitrary automorphism of Pn−1, then λ induces an automorphism on Γ(Pn−1,OPn−1(1)) aswell. This produces a matrix ρ in GLn(E) under the chosen basis. From here we claim directly λ = ρ# since[Har77, Theorem 7.1.(b)] shows any morphism from a variety to Pn−1 is uniquely determined by the imagesof the xi under such a morphism of the global sections.
We start with the notation introduced in the middle row of (D1). The sans-serif font is meant to mean“category”. So SBVn−1(E/k) is a category, whose objects are Severi-Brauer varieties over k, of dimensionn − 1, which split over a Galois field extension E. Such objects can be represented as a pair (X, f) whereX is the variety and f is a choice of splitting of X. The morphisms in this category are the subclass ofmorphisms of varieties which are isomorphisms. (Categories where all morphisms are isomorphisms are calledgroupoids; the category just described is a groupoid). π0SBVn−1(E/k) is the set of isomorphism classes ofthis category. It’s elements can be represented as equivalence classes [(X, f)]. Two objects are isomorphicexactly when the underlying varieties are isomorphic (i.e. two pairs (X, f) and (Y, g) are isomorphic if thereis an isomorphism X ∼= Y ).
Similarly, CSAn(E/k) is a category whose objects are central simple k-algebras, of degree n, which splitover a Galois field extension E. Objects are again pairs, like (A, f), where A is an algebra and f is a choice ofsplitting of A. The morphisms of this category are the subclass of k-algebra morphisms, taking the identityto the identity, which are isomorphisms. π0CSAn(E/k) is the set of isomorphism classes of this category,where two pairs are isomorphic if their underlying algebras are isomorphic.H1(Gal(E/k),PGLn(E)) is the first group cohomology set of Gal(E/k) with values in PGLn(E). This is
just as it is defined in the introductory section so there are no surprises here. The topmost and bottommost13
arrows of the diagram are actually defined on the categorical level, and their description is postponed untilTheorem 4.7. The remaining arrows are described by the next theorem.
Theorem 4.2. There are pointed-bijections
π0SBVn−1(E/k)∼−→ H1(Gal(E/k),PGLn(E))
defined by [(X, f)] 7→ [ϕX,f ], and
π0CSAn(E/k)∼−→ H1(Gal(E/k),PGLn(E))
defined by [(A, f)] 7→ [αA,f ]. More precise definitions of these maps are given in the proof.
Proof. Let G = Gal(E/k). We start with algebras. Let (A, f) be a central simple k-algebra of degree n
with f : A⊗k E∼−→ Mn(E) a splitting of A. Given an element σ of G, there are several actions induced by
σ which we now consider. There is a canonical action of σ on Mn(E), which we write as an isomorphism
σM : Mn(E)∼−→ Mn(E); it is defined by σM ((aij)
ni,j=1) = (σaij)
ni,j=1. There is also a canonical action of σ
on A⊗k E, which we write as an isomorphism σ⊗ : A⊗k E∼−→ A⊗k E; it is defined by σ⊗(a⊗ b) = a⊗ σ(b)
and extended linearly.From these actions we get a square (with all arrows isomorphisms),
A⊗k E Mn(E)
A⊗k E Mn(E)
f
σ⊗ σM
f
which does not commute in general. However, we can correct this lack of commutativity by an automorphismof Mn(E). More precisely, there is a unique element αA,f (σ) of PGLn(E) so that
f σ⊗ = αA,f (σ) · σM f := Int(αA,f (σ))(σM f(−)).
This association of an element σ of G, and the element αA,f (σ) defines a 1-cocycle of Gal(E/k) withvalues in PGLn(E). Said differently, this means αA,f : Gal(E/k) → PGLn(E) is a set map satisfying theadditional condition αA,f (στ) = αA,f (σ) ∗ σαA,f (τ) for all σ, τ in G. This still needs to be checked but, itcan be done:
and, by uniqueness, this means αA,f (στ) equals αA,f (σ) ∗ σαA,f (τ), proving the claim.So far we’ve associated to the pair (A, f) a 1-cocycle of G with values in PGLn(E). We still need
to show this descends to a map on classes, and that the descended map is a bijection. If we show thecohomology class of αA,f is independent of the choice of f , then it follows we have a well-defined mapπ0CSAn(E/k) → H1(G,PGLn(E)). So, let g = ρM f for some ρ in PGLn(E). Then, following theequalities,
g σ⊗ = ρM f σ⊗ = ρ ∗ αA,f (σ) · σM f = ρ ∗ αA,f (σ) · σM (ρ−1M g) = ρ ∗ αA,f (σ) ∗ σρ−1 · σM g
allows us to conclude, again by uniqueness, αA,g(σ) = ρ ∗ αA,f (σ) ∗ σρ−1 which proves the claim.To prove injectivity of the map defined above, suppose [(A, f)] and [(B, g)] are two classes of central
simple algebras giving the same class [αB,g] = [αA,f ] in H1(G,PGLn(E)). We can assume from the startwe have equivalence of 1-cocycles on-the-nose as otherwise, if αB,g 6= αA,f , they differ by an element ρ of
14
PGLn(E) and changing αA,f to αA,ρf doesn’t affect the isomorphism class of pairs. So, we are justified inassuming there exists a diagram
A⊗k E Mn(E) B ⊗k E
A⊗k E Mn(E) B ⊗k E
f
σ⊗ σM
g
σ⊗
f
g
where we have f σ⊗ = αA,f (σ) · σM f and g σ⊗ = αB,g(σ) · σM g. Consequently, we find
f σ⊗ f−1 σ−1M = g σ⊗ g−1 σ−1
M
which implies the outside rectangle of the diagram commutes. Taking G invariants defines an isomorphismA ∼= B, proving injectivity of the map on isomorphism classes.
To prove surjectivity, we start with a given 1-cocycle α and we will construct an algebra A(α) and asplitting f(α) so that αA(α),f(α) = α. To do this, we define a new G-action on the algebra Mn(E): for anyσ of G and x in Mn(E) we let σ · x = α(σ) · σM (x). This is an action since
σ · (τ · x) = α(σ) ∗ σM (α(τ) · τM (x)) = (α(σ) ∗ σα(τ)) · (σM τM (x)) = α(στ)(σM τM (x)) = (στ) · x.We define A(α) := Mn(E)G, the algebra of G-invariant elements under this new action. The map f(α) canbe any map realizing an isomorphism A(α)⊗kE ∼= Mn(E) and adjusted so that we get the required equalityα = αA(α),f(α). This completes the proof except we didn’t check A(α) splits over E. This is the content ofLemma 4.3 below.
The same proof works for the case of varieties. One should note however, the action of G on a varietyX is induced by the opposite action locally. That is to say, if X is affine equal to Spec(A) then the actionof an element σ of G is the morphism σX : X → X corresponding to the canonical ring map σ−1 : A → Ainduced by σ−1. The descent necessary to prove surjectivity is provided by Lemma 4.4.
Lemma 4.3. ([Con, Theorem 2.1.4]) Suppose E is a Galois extension of k with Galois group G = Gal(E/k).Suppose A is a central simple E-algebra (resp. E-vector space, commutative E-algebra) with an action ofG satisfying σ(ax) = σ(a)σ(x) for all σ in G, a in E, and x in A. Then AG is a central simple k-algebra(resp k-vector space, commutative k-algebra). Moreover, there is a canonical isomorphism AG ⊗k E ∼= A ascentral simple E-algebras (resp. E-vector spaces, commutative E-algebras).
Proof. AG is certainly a k-algebra. To see it is central simple, it suffices to prove only AG ⊗k E ∼= A. Theisomorphism given explicitly by the map µ : AG ⊗k E → A on simple tensors µ(x ⊗ a) = ax. In fact, itsuffices to prove this isomorphism only as E-vector spaces, since the given map is an algebra map.
Note, AG 6= 0. This is because there is a k-linear trace map, TrG : A→ AG defined by TrG(x) =∑σ∈G σx;
for a given nonzero element x in A we can consider TrG(ax) =∑σ∈G σ(a)σ(x) for all a in E and, by linear
independence of characters, we can find an a where this is nonzero.To show µ is injective, we take some nonzero element z in AG and expand it in simple tensors z =
∑i xi⊗ai
with the set xi being k-linearly independent. Since µ(z) =∑aixi, to see this is not zero it suffices to
show the xi are also E-linearly independent.So assume we have a relation a1x1 + · · ·+ anxn = 0 for some elements ai in E, with each ai nonzero, and
suppose further we’ve reordered the xi so that such a relation includes a minimal number of the xi. If n = 1,then dividing by a1 shows x1 = 0 and independence follows. If n > 1, then dividing by an we can assumean = 1. Applying an element σ of G to our relation, we find σ(a1)x1 + · · ·+ σ(1)xn = 0 and by subtractingthe two we find (a1 − σa1)x1 + · · ·+ (an−1 − σan−1)xn−1 = 0 which is a relation using strictly less elementsthan the one we started with. By minimality, this implies ai − σai = 0 for each i. Repeating the argumentwe find this is true for all σ in G, and hence each ai is an element of k. Consequently, all xn must be 0 sincethe xn are assumed k-linearly independent, implying the xi are also E-linearly independent.
To show µ is surjective, we can consider the quotient A/µ(AG ⊗k E) as a k[G]-module. The k-lineartrace map TrG descends to a map on the quotient, defined similarly. Considering this as a k-linear mapendomorhpism, the same argument as above shows if there is a nonzero element of A/µ(AG⊗kE) then thereis a nonzero element in
Lemma 4.4. Suppose E is a Galois extension of k with Galois group G = Gal(E/k). Suppose Y is aquasi-projective E-variety with an action of G such that
Y Y
Spec(E) Spec(E)
σY
σE
commutes as a diagram of k-varieties for all σ in G. Then there is a quasi-projective k-variety X and acanonical isomorphism XE = X ×k Spec(E) ∼= Y .
Proof. If Y is affine, equal Spec(A), then the proof of Lemma 4.3 can be applied to A. Otherwise, we canfind a cover of Y by G-invariant affine opens, for each of these affine opens we can apply Lemma 4.3 to geta collection of affine varieties, and finally we can check these glue together to give a variety X satisfying thestatement of the lemma.
Since Y is quasi-projective, there is a locally-closed immersion Y → PnE . We’ll identify Y with the
subvariety of Pn it defines via this immersion. We’ll need one basic fact from algebraic geometry: given aclosed subvariety Z of Pn and a finite number of closed points p1, . . . , pn, there is a hypersurface containingZ but not any of the points p1, . . . , pn.
Pick a closed point y, and let Gy = σY (y)σ∈G denote the G-orbit of y. Using the claim, we can find ahypersurface Hy such that Hy ⊃ Y \ Y and Gy ∩Hy = ∅. By construction, the inclusion
Y \Hy = Y \Hy → Pn \Hy
is a closed immersion. Since closed immersions are affine morphisms and Pn \Hy is affine (if Hy = V+(f)for some polynomial f then Pn \Hy = D+(f) is the basic affine open complement), we find Y \Hy is affineas well. Since the intersection ⋂
σ∈GσY (Y \Hy)
is a finite intersection of affine opens containing y, it is also an affine open containing y; it is G-invariant byconstruction. A G-invariant cover of Y by affine opens is provided by the above procedure as y varies overall closed points of Y .
By another lemma from algebraic geometry (Nike’s trick), we can cover the intersection U ∩V of any twoaffine opens U, V in our cover by affine opens Xii which are basic in both U and V . If we let UG denotethe variety provided by applying Lemma 4.3 to the ring underlying U , then it follows we can glue UG, V G
along the XGi i.
We complete the proof by explaining how one can, given a closed subvariety Z ⊂ Pn and a finite numberof closed points p1, . . . , pn, find a hypersurface Hy in Pn which contains Z and excludes p1, . . . , pn. This isdone using coherent sheaf cohomology.
The subvariety p1 ∪ · · · ∪ pn gives an exact sequence of sheaves on Pn
0→ Ip1∪···∪pn → OPn → Op1∪···∪pn → 0.
Tensoring this sequence by the ideal sheaf of Z gives another exact sequence appearing on the top row ofthe diagram below.
T orOPn
1 (Op1∪···∪pn , IZ) Ip1∪···∪pn ⊗OPnIZ OPn ⊗OPn
IZ Op1∪···∪pn ⊗OPnIZ 0
0 IZ∪p1∪···∪pn IZ Op1∪···∪pn 0
∼ ∼ ∼ ∼
The vertical arrows are isomorphisms (which can be checked on stalks), and so the bottom row is exact aswell. Now, for every integer l we can tensor the bottom row of this diagram by OPn(l), which is an exactOPn -module, to obtain another exact sequence
0→ IZ∪p1∪···∪pn(l)→ IZ(l)→ Op1∪···∪pn(l)→ 0.16
Finally, by general functorality of coherent sheaf cohomology, there is a long exact sequence associated tothis short exact sequence whose first few terms read
By Serre’s vanishing theorem [Har77, Chapter III, Theorem 5.2], H1(Pn, IZ∪p1∪···∪pn(l)) = 0 whenever lis large enough. This implies, for large l, there is a surjection H0(Pn, IZ(l))→ H0(Pn,Op1∪···∪pn(l)). Herethe left group can be identified with homogeneous polynomials “on Pn” vanishing on Z, and the right groupcan be identified with the product of the residue fields of these points, k(p1)×· · ·×k(pn); the map connectingthe two is just the evaluation map. Any polynomial mapping to (1, . . . , 1) will satisfy the necessary propertyof the claim.
A similar statement to Theorem 4.2 holds at the infinite level as well. Given a tower of Galois extensionsE/F/k, there are naturally defined functors SBVn(F/k) → SBVn(E/k) defined on objects by (X, f) 7→(X, f ⊗F E) and by the identity on morphisms. We define SBVn(k) to be the colimit of the system definedby these functors. In a simlar fashion we define CSAn(k) as the colimit of the system defined by functorsCSAn(F/k)→ CSAn(E/k).
Corollary 4.5. There are pointed-bijections
π0SBVn−1(k)∼−→ H1(k,PGLn(ksep))
defined by [(X, f)] 7→ [ϕX,f ], and
π0CSAn(k)∼−→ H1(k,PGLn(ksep))
defined by [(A, f)] 7→ [αA,f ].
Proof. By definition, H1(k,PGLn(ksep)) is the profinite cohomology of the profinite group Gk with values inthe continuous Gk-group PGLn(ksep). That this latter object is in fact a Gk-group can be checked straightfrom the definition (any element x of PGLn(ksep) is represented by a matirx with finitely many coefficients;adjoining all of these coefficients to k gives a finite extension E and the stabilizer of x can be identified withGal(ksep/E); the compatibly of action and group operation is immediate).
By Theorem 4.2 the horizontal arrows in the diagram below are isomorphisms.
π0SBVn−1(F/k) H1(Gal(F/k),PGLn(F ))
π0SBVn−1(E/k) H1(Gal(E/k),PGLn(E))
The left vertical arrow is induced by the maps of the discussion prior to the corollary, and the right verticalarrow is an inflation map of group cohomology. The diagram commutes so, taking colimits of the systemsdefined by the vertical maps as E varies over all finite Galois extensions of k gives the result.
Remark 4.6. For any field E and any n,m > 0, there are maps PGLn × PGLm → PGLnm given by thetensor product ([x], [y]) 7→ [x⊗ y]. When E is a Galois extension of k with Galois group G, this defines aninductive system of maps on cohomology
In particular, when E is a Galois extension of k with Galois group G = Gal(E/k), Proposition 1.14 givesrise to an exact sequence which ends with a boundary map
∂n : H1(G,PGLn(E))→ H2(G,E×).17
These boundary maps are compatible with the inductive system defined above, as n varies over all positiveintegers. All together, we’ve defined a map
lim−→n
∂n : lim−→n
H1(G,PGLn(E))→ H2(G,E×)
which is, in fact, an isomorphism [GS06, Lemma 4.4.4].In a similar fashion, for any Galois extension E of k and any n,m > 0, there are canonical maps
π0CSAn(E/k) × π0CSAm(E/k) → π0CSAnm(E/k) given by the tensor product of central simple k-algebras[(A, f)], [(B, g)] 7→ [(A⊗k B, f ⊗ g)]. This defines an inductive system of maps
which is evidently compatibile under the correspondence π0CSAn(E/k) ∼= H1(Gal(E/k),PGLn(E)) of The-orem 4.2 with the system defined in the paragraph before. Thus, we have described an isomorphismlim−→n
π0CSAn(E/k) ∼= lim−→nH1(G,PGLn(E)).
The maps PGLn(E) × PGLm(E) → PGLnm(E) can be interpreted as defining a product on the col-imit lim−→n
H1(G,PGLn(E)) ∼= lim−→nπ0CSAn(E/k). The isomorphism lim−→n
∂n is an isomorphism of groups
when the domain is equipped with this operation. Working through the description of this product onlim−→n
π0CSAn(E/k) it turns out we have described an isomorphism
H2(G,E×) ∼= lim−→n
H1(G,PGLn(E)) ∼= lim−→n
π0CSAn(E/k) ∼= Br(E/k).
In particular, this is the reason for an equality in Remark 2.6. It, moreover, explains why the map δhas finite image: group cohomology of a finite group G is torsion in degrees greater or equal 1. Takingcolimits over Galois extensions E/k, we can also obtain an isomorphism Br(k) ∼= H2(k,Gm(ksep)). This isthe cohomological interpretation of the Brauer group.
Remark 4.7. This remark describes an efficient way to check whether a Severi-Brauer curve is split or not;this is called the Hilbert symbol.
From the exact sequence
1→ µ2 → Gm(ksep)→ Gm(ksep)→ 1
one gets a long exact sequence in Galois cohomology. Part of this exact sequence reads
1→ k×sep×2−−→ k×sep → H1(k, µ2)→ H1(k,Gm(ksep)).
The group H1(k,Gm(ksep)) is trivial by Hilbert’s Theorem 90, Lemma 4.10 below, so we get an isomorphismk×/k×2 ∼= H1(k, µ2) whenever char(k) 6= 2. Now, one can consider the tensor product and the cup productcomposition
k× ⊗Z k× → H1(k, µ2)⊗H1(k, µ2)
a⊗b 7→a∪b−−−−−−→ H2(k, µ2 ⊗Z µ2) ∼= H2(k, µ2).
This map factors through Milnor K-theory mod 2 to give an isomorphism
KM2 (k)/2
∼−→ H2(k, µ2).
Here the Milnor K-theory group KM2 (k) is defined as the quotient of the group k× ⊗Z k
× by the relationsa⊗ (1− a) for all a 6= 1 and this isomorphism is a specific case of the Merkurjev-Suslin isomorphism, [GS06,Theorem 8.6.5].
From the commutative diagram
1 µ2 SL2(ksep) PGL2(ksep) 1
1 Gm(ksep) GL2(ksep) PGL2(ksep) 1
18
we get a commuting square
H1(k,PGL2(ksep)) H2(k, µ2)
H1(k,PGL2(ksep)) H2(k,Gm(ksep))
and the top vertical arrow is an inejection because the bottom-horizontal and left-vertical maps are injective.7
Composing with the canonical bijection of Corollary 4.5 π0SBV1(k)→ H1(k,PGL2(ksep)) and the Merkurjev-Suslin isomorphism, one gets a well-defined function π0SBV1(k) → KM
2 (k)/2 which takes the class of P1k
to the identity element of KM2 (k)/2. Denoting the image of a pure tensor a ⊗ b in KM
2 (k)/2 by a symbola, b, this function is defined by taking the class of the Severi-Brauer curve defined by the polynomialax2 + by2 = z2 to the symbol a, b. Now one can use the group structure of KM
2 (k)/2 to determine whetheror not a symbol a, b is trivial instead of determining whether or not ax2 + by2 = z2 is split.8
Although the relationship between isomorphism class of Severi-Brauer varieties, isomorphism classes ofcentral simple algebras, and group (or Galois) cohomology is helpful for computational purposes, it’s oftenbeneficial to know how to go from an algebra to a variety (or vice versa) more explicitly. In one direction,we start with an arbitrary central simple k-algebra A of degree n and aim to describe a natural processobtaining a Severi-Brauer k-variety SB(A).
As A is a k-vector space, we can consider the Grassmannian of n-dimensional subspaces contained inA, Gr(n,A). There is a subvariety of Gr(n,A) whose points are those subspaces which are left ideals ofA. This is easier seen using the Plucker embedding of the Grassmannian Gr(n,A) → P (
∧nA). The
Plucker embedding is given by showing the set of irreducible wedges cut out a variety in P(∧n
A); see[Mil17a, Proposition 6.29, Lemma 6.35] for more details. Similarly, the embedding we want is describableby polynomials which characterize the condition of being a left ideal in A. We fix a k-basis e1, . . . , en2
for A. Given a n-dimensional space corresponding to a point u1 ∧ · · · ∧ un of P(∧n
A), the conditionW = u1 · k + · · · + un · k is an ideal corresponds to the conditions e1W ⊂ W, . . . , en2W ⊂ W . This isequivalent to the conditions (eiu1 ∧ · · · ∧ eiun) ∧ uj = 0 for all 1 ≤ i ≤ n2 and all 1 ≤ j ≤ n. As with thePlucker embedding, this can be translated into a system I of polynomial constraints. We define SB(A) tobe the subvariety of Gr(n,A) described by the vanishing of the polynomials in I.
When A = Mr(k), we have an isomorphism A ∼= V ⊗k V ∨ where V = k⊕n. This lets us define a morphismP(V ∨) → Gr(n, V ⊗ V ∨) given on points by W 7→ V ⊗k W . This map turns out to be an isomorphismwith the subvariety SB(A), which can be checked locally on a choice of affine opens; see [KMRT98, Theorem1.18] but, note our conventions are the opposite of the ones in this source. The choice of splitting f for anarbitrary central simple algebra A then corresponds to the choice of a splitting for the Severi-Brauer varietySB(A), and we denote this choice by SB(f).
We define the functor SB : CSAn(E/k) → SBVn−1(E/k) by setting (A, f) 7→ (SB(A),SB(f)) on objectsand by the canonical maps on morphisms.
Conversely, let X be a Severi-Brauer k-variety. We can associate to X a central simple algebra using someinformation on the finite rank locally free sheaves on X. If X = Pn
k , then there is an exact sequence
0→ Ω1Pn → OPn(−1)⊕n+1 → OPn → 0
called the Euler exact sequence. Using Yoneda-extensions, this sequence corresponds to an element in theExt∗OPn
(OPn ,OPn) ' k-vector space Ext1OPn
(OPn ,Ω1Pn) ' H1(Pn,Ω1
Pn) ∼= k. The endomorphism ring of
7The right-vertical map in this diagram is also injective and identifies H2(k, µ2) with the 2-torsion of H2(k,Gm(ksep)). Tosee this, apply the long exact cohomology sequence to
1→ µ2 → Gm(ksep)→ Gm(ksep)→ 1
to get the exact sequenceH1(k,Gm(ksep))→ H2(k, µ2)→ H2(k,Gm(ksep))
and then apply Hilbert’s Theorem 90, Lemma 4.10 below, to the leftmost group.8Here are some examples to illustrate this. We’ve determined in Example 2.4 the symbol −1,−1 in KM
2 (R)/2 is nontrivial.Then, whenever a, b > 0, the symbol −a,−b is nontrivial as well. This is just the bilinearity relation −a,−b = a,−b +
−1,−b = a,−b+ −1,−1+ −1, b and the fact −b, a = −1, b are both trivial (since a, b > 0, they are squares in R,
so these symbols are 0 in the mod 2 Milnor K-group).19
the sheaf in the middle of this sequence is isomorphic to a matrix ring, End(⊕n
i=0OPn(−1)) ∼= Mn+1(k);this can be realized by the natural isomorphisms
End
(n⊕i=0
OPn(−1)
)=Hom
n⊕i=0
OPn(−1),
n⊕j=0
OPn(−1)
'
n⊕i,j=0
Hom(OPn(−1),OPn(−1))
'n⊕
i,j=0
Hom(OPn ,OPn) ∼= Mn+1(k)
and a choice of basis to get the last isomorphism.For an arbitrary Severi-Brauer k-variety X, we mimic this process. The crucial observation is that there
is a uniquely determined non-split extension
0→ Ω1X → EX → OX → 0
which upon base change to a splitting field E of X, becomes isomorphic to the Euler exact sequence onprojective space over E. If we show the Ext∗OX (OX ,OX) ' k-vector space Ext1
OX (OX ,Ω1X) ' H1(X,Ω1
X) isisomorphic with k, we can define this sequence to be any exact sequence which defines, via Yoneda-extensionclasses, a nonzero element η in Ext1
OX (OX ,Ω1X). However, flat base change gives an isomorphism
E ∼= H1(XE ,Ω1XE ) ' H1(X,Ω1
X)⊗k E
and therefore the desired isomorphism H1(X,Ω1X) ∼= k. As before, a choice of spltiting f determines a
splitting of the corresponding algebra, which we denote by AZ(f).We define the functor AZ : SBVn−1(E/k) → CSAn(E/k) by setting (X, f) 7→ (EndOX (EX),AZ(f)) on
objects and by the canonical maps on morphisms.
Theorem 4.8. For any field extension E of k there are mutually inverse equivalences
SB : CSAn(E/k) SBVn−1(E/k) : AZ
that induce maps π0SB and π0AZ which make the diagram (D1) commutative.
Reference. [Jah00, Theorem 6.19].
Finally, we end by completing the proof of Proposition 2.8.
Theorem 4.9. If a Severi-Brauer k-variety X of dimension n contains a k-rational point, then X is iso-morphic to Pn
k .
Proof. Let p be the k-rational point of X, E a Galois extension of k which splits X, and f : X ×k E∼−→ Pn
E
a splitting of X. Let q = p ×k E be the E-rational point of X ×k E defined by p. Changing f by anautomorphism of Pn
E if necessary, we can assume f(q) = (1 : 0 : · · · : 0).The pair (X, f) gives rise to a 1-cocycle αX,f by the procedure explained in the proof of Theorem 4.2.
We’ll show αX,f is cohomologous to the trivial 1-cocycle αPnk ,idEdefined by (Pn
k , idE). By Theorem 4.2, thisimplies the two varieties X and Pn
k are isomorphic, which will complete the proof.To start, observe αX,f (σ)(f(q)) = f(q) for all σ in G = Gal(E/k). This is because αX,f (σ) is defined as
the unique map so that f σXE = αX,f (σ) · σPn f but, in our situation,
That σXE (q) = q is best seen locally: choosing an open affine around q we can assume q is given by an idealin a ring isomorphic with k[x0, . . . , xs]/(f1, . . . , ft)⊗k E; by Hilbert’s weak Nullstellensatz applied to p, weknow q is of the form (x0 ⊗ 1 − a1 ⊗ 1, . . . , xs ⊗ 1 − as ⊗ 1) and the action of σ is then visibly trivial as itacts only on the second component of a tensor.
20
Let P be the subgroup of PGLn(E) which fixes (1 : 0 : · · · : 0). Since the action of PGLn(E) on PnE is
induced by the action of GLn(E), we can actually identify P as the image of the subgroup
under the natural quotient map GLn(E)→ PGLn(E). Note, however, there is a subgroup of S
P ′ = (aij)ni,j=1 ∈ S : a1,1 = 1
which also maps surjectively to P . Moreover, there is trivial intersection P ′ ∩Gm = 1 between P ′ and thecenter E× = Gm(E) ∼= Z(GLn(E)) of GLn(E). This implies there is the following commutative diagram,where the rows are exact and the vertical maps are inclusions.
1 P ′ ∩Gm = 1 P ′ P 1
1 Gm(E) GLn(E) PGLn(E) 1
∼
Applying Proposition 1.8 to this diagram we get the following commuting square.
H1(G,P ′) H1(G,P )
H1(G,GLn(E)) H1(G,PGLn(E))
∼
The class [αX,f ] is in the image of the right vertical map by assumption. By commutativity of this diagram,this means it is also in the image of the bottom horizontal arrow. Then, by Lemma 4.11 below and thefact these maps are maps of pointed sets, we find [αX,f ] = [αPnk ,idE
]; this completes the proof as explainedabove.
Lemma 4.10. (Hilbert’s Theorem 90)9 Let E be a Galois extension of k with Galois group G = Gal(E/k).For all n ≥ 0, the group GLn(E) is canonically a G-group and, with its canonical G-group structure, thefirst cohomology of G with coefficients in GLn(E) is trivial: H1(G,GLn(E)) = 1.
Proof. Suppose we’re given an arbitrary 1-cocycle α : G→ GLn(E). The argument in the proof of surjectivityin Theorem 4.2, and the Galois descent from Lemma 4.3, can be applied to show α is the 1-cocycle associatedto a twisted form V of the E-vector space En. As vector spaces, all twisted forms are trivial: if we have ak-vector space V with V ⊗k E ∼= En then V ∼= kn (the trivial form) because V has dimension n.
References
[Art82] M. Artin, Brauer-Severi varieties, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), LectureNotes in Math., vol. 917, Springer, Berlin-New York, 1982, pp. 194–210. MR 657430
[Bru] E. Brussel, Galois descent and Severi-Brauer varieties, Unpublished notes, http://www.mathcs.emory.edu/
~brussel/Papers/galoisdescent.pdf.[CM06] V. Chernousov and A. Merkurjev, Connectedness of classes of fields and zero-cycles on projective homogeneous
[Con] K. Conrad, Galois descent, Unpublished notes, http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/
galoisdescent.pdf.
[FGI+05] B. Fantechi, L. Gottsche, L. Illusie, S. L. Kleiman, N. Nitsure, and A. Vistoli, Fundamental algebraic geom-etry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, RI, 2005,
Grothendieck’s FGA explained. MR 2222646
[GS06] P. Gille and T. Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathe-matics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528
[GW10] U. Gortz and T. Wedhorn, Algebraic geometry I, Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden,
2010, Schemes with examples and exercises. MR 2675155
9This statement is one generalization of Hilbert’s original Theorem 90. In its original formulation, Hilbert’s theorem gives
an explicit description of a basis for a cyclic Galois extension E/k as a k-vector space. In today’s language, it would be easier
to say Hilbert showed E was a trivial k[G]-module where G = Gal(E/k). By some cohomological facts, this is equivalent tosaying H1(G,E×) = 0 when E and G are as above. The theorem generalizes this to arbitrary Galois extensions (not necessarily
cyclic ones) and to a larger class of coefficients (to all GLn instead of just n = 1).
[Har77] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, Graduate Texts in Mathematics,No. 52. MR 0463157
[Jac96] N. Jacobson, Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, 1996. MR 1439248
[Jah00] J. Jahnel, The Brauer-Severi variety associated with a central simple algebra: a survey, Unpublished notes, https://www.math.uni-bielefeld.de/LAG/man/052.pdf, 2000.
[Kar98] N. A. Karpenko, Codimension 2 cycles on Severi-Brauer varieties, K-Theory 13 (1998), no. 4, 305–330. MR 1615533
[KMRT98] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical SocietyColloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998, With a preface in French
by J. Tits. MR 1632779[Kol16] J. Kollar, Severi-Brauer varieties; a geometric treatment, ArXiv e-prints (2016), https://arxiv.org/abs/1606.
04368.
[Kun13] E. Kunz, Introduction to commutative algebra and algebraic geometry, Modern Birkhauser Classics,Birkhauser/Springer, New York, 2013, Translated from the 1980 German original [MR0562105] by Michael Acker-
man, With a preface by David Mumford, Reprint of the 1985 edition [MR0789602]. MR 2977456
[Liu02] Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford UniversityPress, Oxford, 2002, Translated from the French by Reinie Erne, Oxford Science Publications. MR 1917232
[Mil80] J. S. Milne, etale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J.,
1980. MR 559531[Mil13a] , Class field theory (v4.02), 2013, Available at www.jmilne.org/math/, pp. 281+viii.
[Mil13b] , Lectures on etale cohomology (v2.21), 2013, Available at www.jmilne.org/math/, p. 202.[Mil17a] , Algebraic geometry (v6.02), 2017, Available at www.jmilne.org/math/, p. 221.
[Mil17b] James S. Milne, Fields and galois theory (v4.53), 2017, Available at www.jmilne.org/math/, p. 138.
[Ser97] J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997, Translated from the French by Patrick Ion and revisedby the author. MR 1466966
[Sta17] The Stacks Project Authors, exitStacks Project, http://stacks.math.columbia.edu, 2017.
[Vis08] A. Vistoli, Notes on grothendieck topologies, fibered categories, and descent theory, Unpublished notes, http://
homepage.sns.it/vistoli/descent.pdf, 2008.
Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA
