Examples of Brauer-Severi schemes

(1)

MSc Mathematics

Master Thesis

Examples of Brauer-Severi Schemes

Author: Supervisor:

Zo¨

e Schroot

Prof. dr. Lenny Taelman

Examination date:

(2)

Abstract

The goal of this thesis is to construct explicit examples of non-linear Brauer-Severi schemes. First, we prove that on a smooth projective surface where the positive integer n is invertible, there exist non-linear Brauer-Severi schemes of rank n − 1 over S iff the Brauer group has non-trivial n-torsion. Then we construct examples of rank 1 Brauer-Severi schemes from two elements of H1(S´et, µ2,S), by making a construction analogous

to the quaternion algebra-conic correspondence. Moreover, we prove that the class in the Brauer group of the constructed Brauer-Severi scheme is given by the cup product of the elements it is built of. Finally, we deduce that each non-split short exact sequence of finite groups contained in 1 → Gm,S → GLn,S → PGLn,S → 1 gives rise to a construction

of Brauer-Severi schemes.

Title: Examples of Brauer-Severi Schemes

Author: Zo¨e Schroot, zoe.schroot@student.uva.nl, 10399437 Supervisor: prof. dr. Lenny Taelman

Second Examiner: dr. Mingmin Shen Examination date: June 11, 2018

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

(3)

Introduction

Elements of the Brauer group arise naturally as obstructions to the fineness of moduli spaces, or as obstruction for the existence of rational points (see e.g. [Aue+17]). Accord-ingly, it is essential to characterise the Brauer group and its elements. There are two equivalent descriptions of the elements of the Brauer group of a scheme, as Azumaya algebras or Brauer-Severi schemes [Gro95]. In this theris we will focus on Brauer-Severi schemes. The aim is to construct explicit examples of Brauer-Severi schemes defining non-trivial elements in the Brauer group.

Brauer-Severi schemes are defined as ´etale locally trivial projective bundles [Gro95, rk. 8.1]. Usually they are described using gluing data, which is useful, but doesn’t provide the full picture. Therefore, we will look for global descriptions of Brauer-Severi schemes.

The obvious examples of Brauer-Severi schemes are projectivization of vector bundles, which are precisely the schemes defining the trivial class in the Brauer group. We will call these kind of Severi schemes linear. We are interested in finding Brauer-Severi schemes not of this form, the non-linear Brauer-Brauer-Severi schemes. Note that over an algebraically closed field, all Brauer-Severi schemes are linear (see e.g. [GS17]). Moreover, by Tsen’s theorem ([Stacks, Tag 03RF]), all Brauer-Severi schemes on a curve over an algebraically closed field are linear.

This raises the question whether the same is true for surfaces over an algebraically closed field. It turns out that this is indeed the case for smooth projective surfaces of Kodaira dimension -1, since the Brauer group is trivial [DF84, Cor. 6]. On the other hand, the Brauer group of a K3 surface X over C is given by (Q/Z)⊕22−ρ(X), where ρ(X) is the Picard number [Huy16, Ch.18 1.2]. This implies that there exist non-linear Brauer-Severi schemes over a K3 surface, but it is not clear of which rank.

We will use de Jong’s result about the period-index problem for the function field of a surface over a separably closed field [Jon04], to prove that there exist non-linear Brauer-Severi schemes of rank n−1 over S if and only if the Brauer group has non-trivial n-torsion. The precise statement is:

Proposition 0.1. Let S be a smooth projective surface over an algebraically closed field k and n ∈ Z>0 with char(k) - n, then the inclusion δn: H1(S´et, PGLn,S) → Br(S) surjects

onto the n-torsion of Br(S). Moreover, it fits into an exact sequence H1(S´et, GLn,S) → H1(S´et, PGLn,S) → Br(S)[n] → 1.

In particular, there exist surfaces with non-trivial n torsion. So this result gives that there exist non-linear rank n − 1 Brauer-Severi schemes over these surfaces.

(5)

For example, a K3 surfaces over C has n22−ρ(X) different n-torsion elements in the Brauer group. Using that only the trivial Brauer class corresponds to linear Brauer-Severi schemes, this gives that there exist at least n22−ρ(X)− 1 isomorphism classes of Brauer-Severi schemes of rank n − 1 over a K3 surface. Unfortunately, the proof the proposition is not constructive.

The rest of the thesis is devoted to the construction of non-linear Brauer-Severi schemes. Besides the standard example, the universal conic, there are not many ex-plicit descriptions of non-linear Brauer-Severi schemes known. Since we have seen there exist non-linear Brauer-Severi schemes over a surface, we will start with introducing a modification of this standard example, the universal diagonal conic. This turns out the be a non-linear Brauer-Severi scheme of rank 1 over an affine surface. This provides the basis for all other constructions.

We start with an explicit construction for Brauer-Severi schemes of rank 1 over any base scheme. For this construction, the main observation is that we can gener-alise the assignment of a quaternion algebra (a, b)k to two elements a, b ∈ k∗/(k∗)2 ∼=

H1(Spec(k)´et, µ2,S), and the correspondence between quaternion algebras and conics.

For any scheme S we built an Azumaya OS-algebra (α, β)S of degree 2 from two

el-ements (L, α), (M, β) ∈ H1(S´et, µ2,S), here L, M ∈ Pic(S) and α : L⊗2 → OS and

β : M⊗2 → O_S are trivialisations of the second tensor power. Moreover, to this Azu-maya algebra we can associate a conic bundle, which is a Brauer-Severi scheme of rank 1. Furthermore, just as the class of (a, b)kin the Brauer group is given by the cup product

([GS17, Prop. 4.7.1]), we prove that the class of (α, β)S is given by the cup product:

Proposition 0.2. Let S be a scheme such that 2 ∈ O_S∗ and take (L, α), (M, β) ∈ H1(S´et, µ2,S), then (α, β)S = α ∪ β ∈ H2(S´et, Gm,S).

This provides us with an easy way to check whether the corresponding Brauer-Severi scheme is non-linear, precisely if the cup product is non-trivial.

The proof of this proposition reveals that this construction arises from the short exact sequence of finite groups 1 → Z/2Z → D4 → Z/2Z × Z/2Z → 1 contained in

1 → Gm,S → GLn,S → PGLn,S → 1 forming the commuting diagram

1 µ2,S D4 µ2,S× µ2,S 1

1 Gm,S GL2,S PGL2,S 1.

Using the Kummer sequence we get that this short exact sequence of finite groups induces possibly non-linear Brauer-Severi schemes because it is non-split and the cup product is in general non-trivial.

In general, every non-split short exact sequence of constant finite group schemes 1 → µm,S → HS → GS → 1 contained in 1 → Gm → GLn → PGLn → 1 gives

rise to a construction of Brauer-Severi schemes. We need non-split because otherwise the constructed Brauer-Severi schemes are all linear. Precisely, the Kummer sequence gives that in order to get non-linear Brauer-Severi schemes one needs that the image

(6)

of δG: H1(S´et, GS) → H2(S´et, µm,S) is not contained in the image of Pic(S) ⊗ Z/mZ in

H2(S´et, µm,S). Using that an element T ∈ H1(S´et, GS) is representable by a GS-cover

T , we get that the induced Brauer-Severi scheme is given by (T × Pn−1S )/GS. Thus,

all short exact sequences as above give us a construction of non-linear Brauer-Severi schemes, with an elegant, explicit description. Moreover, this provides examples of (non-linear) Brauer-Severi schemes over any base scheme and rank.

Outline

In chapter 1 we will recall the basics of ´etale topology and sheaves on a site in order to be able to define the Brauer-Severi schemes and Azumaya algebras. Then we will review some results on non-abelian sheaf cohomology, which provide the language and tools to show the connection between Azumaya algebras and Brauer-Severi schemes. We will conclude this chapter with the essential short exact sequences and their implication on the level of cohomology, which will be needed to prove 0.1.

In chapter 2 we will introduce the main objects, Brauer-Severi schemes, Azumaya algebras and their connection. Then, we will look at Brauer group and the period-index problem. In particular, the theorems known for central simple algebras, such as Tsen’s theorem and de Jong’s theorem, and when we can apply these to Azumaya algebras. In the last section everything comes together. Using de Jong’s theorem and the sequences of chapter 1, we prove proposition 0.1 about the existence of non-linear rank n − 1 Brauer-Severi schemes.

Chapter 3 is devoted to the construction of non-linear Brauer-Severi schemes. We start with an example; the universal diagonal conic. Then, we will give a construction from double covers, and give the corresponding class in the Brauer group by proving 0.2. Finally, we give a general construction from non-split short exact sequences of finite groups.

(7)

1 Preliminaries

As stated in the introduction, this thesis is devoted to the construction of Brauer-Severi schemes. In order to define the Brauer-Severi schemes, we need to recall the ´etale topology and sheaves in this topology. Then we will introduce the tools which will allow us to prove that non-linear Brauer-Severi schemes exist; non-abelian sheaf cohomology, the Kummer sequence and the projective linear group on a scheme PGLn,S.

1.1 The ´

etale topology

The étale topology is a Grothendieck topology, which is the part of a topology needed to define sheaves. It is the topology one gets by resolving the failure of the inverse function theorem in the Zariski topology. Its main use is the associated cohomology theory: étale cohomology. The étale cohomology of complex varieties with values in finite fields give the usual Betti numbers [Stacks, Tag 03N7], so in this sense the étale topology behaves more like the analytic topology. Because of its nice properties étale topology is widely used.

Let us start by defining a Grothendieck topology on a category.

Definition 1.1 ([Stacks, Tag 00VG]). Let C be a category. A Grothendieck topology on C is the assignment to each object U of C of a collection of sets of arrows {Ui → U },

called coverings of U , so that the following conditions are satisfied; (i) if V → U is an isomorphism, then the set {V → U } is a covering,

(ii) if {Ui → U } is a covering and V → U is any arrow, then the fibered products

{U_i×_U V } exist and the collection of projections {Ui×UV → V } is a covering,

(iii) if {Ui → U } is a covering and for each index i we have a covering {Vij → Ui}, the

collection of composites {Vij → Ui→ U } is a covering of U .

A category with a Grothendieck topology is called a site.

Example 1.2. (The classical site of a topological space) Let X be a topological space and denote by Xclthe category in which the objects are the open subsets of X and morphisms

given by inclusions. We can define a Grothendieck topology on Xcl by assigning to each

object U of Xcl the collection of open coverings of U . Note that in this case for arrows

U1 → U and U2 → U the fibered product U1×U U2 is the intersection U1∩ U2.

Example 1.3. (The global classical topology) Similar to the previous example we can define a Grothendieck topology on the category Top of topological spaces. This time, if U is a topological space, a covering of U is a jointly surjective collection of open embeddings Ui → U .

(8)

The ´etale topology is defined using ´etale morphsims, which is the analogue of local homeomeorphisms for topological spaces.

Definition 1.4. ([Stacks, Tag 02GH]). Let R be a ring and h, g ∈ R[x] where g is monic and its derivative g0 is invertible in Rh[x]/(g). Then the ring map R → Rh[x]/(g) is said

to be standard ´etale.

A morphism of affine schemes f : X → S is called standard ´etale if X → S is isomorphic to

Spec(Rh[x]/(g)) → Spec(R),

where R → Rh[x]/(g) is a standard ´etale ring map.

A morphism of schemes f : X → S is ´etale if there exists an open affine covering {Ui}

of X and open affines {Vi} in S such that f (Ui) ⊂ Vi and f |Ui: Ui → Vi is standard

´etale.

Example 1.5. Let k be a field, then a standard ´etale morphism corresponds to a product of finite separable field extensions: For g ∈ k[X] we have g0 ∈ (k[X]/(g))∗ _{if and only if}

gcd(g, g0) = 1, which implies that g has different roots and hence separable. Using the Chinese remainder theorem on the irreducible components of g gives that k[X]/(g) is a finite product of finite separable field extensions.

Example 1.6. ([Stacks, Tag 03N4]). On smooth projective varieties over C, an ´etale morphism is a local homeomorphism in the analytic topology.

Properties 1.7. ([Stacks, Tag 02GH]). Some properties of étale morphisms are; (i) the composition of two étale morphisms is étale,

(ii) the base change of a morphism which is étale is étale, (iii) any open immersion is étale,

(iv) an ´etale morphism is open,

(v) an ´etale morphism is locally of finite presentation,

(vi) let f : X → Y be a morphism of schemes over S. If X and Y are ´etale over S, then f is ´etale.

Definition 1.8 ([Stacks, Tag 0214]). (The small étale site of a scheme) Let S be a scheme. Consider the full subcategory Sét of (Sch/S), consisting of étale morphisms

U → S. Note that if U → S and V → S are objects of S´et, then an arrow U → V is

´etale by 1.7. We define a covering of U → S ∈ S´etto be a jointly surjective collection of

morphisms Ui → U .

Definition 1.9 ([Stacks, Tag 0214]). (The global étale topology) We define the étale topology on (Sch/S) by defining a covering of an object U → S of (Sch/S) to be a jointly surjective collection of étale morphisms {Ui → U }.

(9)

Grothendieck topologies are constructed in order to be able to define sheaves. Recall that for any topological space X, a presheaf of sets on X is a functor X_clop→ Sets. The sheaf condition can now be generalised to any site when we substitute intersections with fibered products. Moreover, each presheaf has a sheafification.

Definition 1.10 ([Stacks, Tag 00VL]). Let C be a site and F : Cop→ Sets a functor. The functor F is a sheaf if the following condition is satisfied. Given a covering {Ui→ U } in

C and a set of elements ai ∈ F (Ui). Let pr1: Ui×UUj → Ui and pr2: Ui×UUj → Uj be

the first and second projection respectively and assume that pr₁∗ai= pr2∗aj ∈ F (Ui×UUj)

for all i and j. Then there exists a unique section a ∈ F (U ) whose pullback to F (Ui) is

ai for all i.

If F and G are sheaves on a site C, a morphism of sheaves F → G is a natural transformation of functors.

One can also define sheaves of groups, rings etc. A functor from Cop to the category groups or rings, is a sheaf if its composite with the forgetful functor to the category of sets is a sheaf.

Remark. Note that for a topological space X, a sheaf over Xcl is a precisely sheaf in the

usual sense.

Definition 1.11 ([Stacks, Tag 00W1]). Let C be a site and F : C → Sets a functor. A sheafification of F is a sheaf Fa: C → Sets, together with a natural transformation F → Fa, such that;

(i) given an object U of C and ξ, η ∈ F (U ) whose images ξa and ηa in Fa(U ) are isomorphic, there exists a covering {σi: Ui → U } such that σ_i∗ξ = σ_i∗η, and

(ii) for each object U of C and each ¯ξ ∈ Fa(U ), there exists a covering {σi: Ui → U }

and elements ξi ∈ F (Ui) such that ξia= σ∗iξ.¯

Example 1.12 ([Stacks, Tag 03YZ, Tag 047F]). Let S be a scheme, we will give a few examples of sheaves over S´et:

(i) The structure sheaf OS is the functor sending (U → S) ∈ S´et to Γ(U, OU).

(ii) The sheaf Gm,S is the functor sending (U → S) ∈ S´etto Γ(U, O_U∗).

(iii) The sheaf µn,S is the functor sending (U → S) ∈ S´et to µn(U ) := {f ∈ Γ(U, O∗U) |

fn= 1}.

(iv) The sheaf GLn,S is the functor sending (U → S) ∈ S´et to GLn(OS(U )).

(v) The constant sheaf Z/nZ is the sheafification of the constant presheaf sending (U → S) ∈ S´et to Z/nZ.

(10)

Definition 1.13 ([Stacks, Tag 06UM]). The global sections of a (pre)sheaf of sets F on a site C is the set

Γ(C, F ) := M orPSh(C)(∗, F ),

where ∗ is the final object in the category of presheaves on C, i.e. the presheaf which associates to every object a singleton.

Since the category of sheaves of abelian groups on a site has enough injectives ([Stacks, Tag 01DP]), we can define sheaf cohomology on a site as the right derived functors of the global section functor, analogously to sheaf cohomology in the Zariski topology, see [Stacks, Tag 01FT]. For the étale site, this gives us étale cohomology, which has become a widely used tool in modern algebraic geometry. For more details about étale cohomology we refer to [Mil80] and [Mil98].

1.2 Non-abelian sheaf cohomology

In this section we will give the basics of non-abelian sheaf cohomology and torsors, as this framework will allow us to construct a correspondence between Azumaya algebras and Brauer-Severi schemes. We will illustrate the idea of this correspondence by look-ing at the correspondence between isomorphism classes of line bundles and cocycles in H1(S, O∗_S) and show how we can use non-abelian sheaf cohomology to generalise this to a correspondence of isomorphism classes of vector bundles with H1_{(S, GL}

n,S). As in

these examples, only the first cohomology set is of interest to us. Because this is defined using torsors, we start of with the definition of a torsor.

Definition 1.14. ([Stacks, Tag 03AG]). Let C be a site and G ∈ Grp(C) a sheaf of groups. A left pseudo G-torsor on C is a sheaf of sets F endowed with an action G × F → F such that the action G(U ) × F (U ) → F (U ) is free and transitive whenever F (U ) is non-empty.

A morphism of pseudo G-torsors is a morphism of sheaves compatible with the group action.

A pseudo G-torsor is a G-torsor when for every object U in C there exists a covering {U_i → U }_i∈I such that F (Ui) is non empty for all i ∈ I. Note that when F (Ui) has a

section, it is isomorphic to G(Ui) since G(Ui) × F (Ui) → F (Ui) is free and transitive.

A morphism of G-torsors is a morphism of pseudo G-torsors.

Remark. In differential geometry torsors correspond to principal bundles.

Construction 1.15 ([Gir71, Ch.3 Cor. 2.2.6]). (The Isom construction.) Let C be a site. There is an easy way to construct a torsor from two locally isomorphic sheaves.

Let F1, F2 be sheaves on C. Define the sheaf Isom(F1, F2) as Isom(F1, F2)(U ) :=

Isom(F1|U, F2|U) the sheaf isomorphims and Aut(F ) := Isom(F , F ). Then there is a

natural left action of Aut(F2) on Isom(F1, F2) given by

(11)

This action is free and transitive, so when F1, F2 are locally isomorphic, Isom(F1, F2)

becomes a left Aut(F2)-torsor. Similarly, Isom(F1, F2) becomes a right Aut(F1)-torsor,

so Isom(F1, F2) is actually a bi-torsor.

Example 1.16. Let (X, OX) be a ringed space and L a line bundle. Then Aut(OX) =

O∗_X and therefore Isom(L, OX) is a left O∗_X-torsor.

Similarly, let V be a rank n vector bundle on X. Then Aut(O⊕n_X ) = GLn,X and

therefore Isom(V, O⊕n_X ) defines an GLn,X-torsor.

Construction 1.17 ([Gir71, Ch.3 Def. 1.3.1]). (The twist construction.) Let G ∈ Grp(C), T a G-torsor and F be a sheaf with a right G action. Define the diagonal group action F ×T ×G → F ×T by (x, y, g) 7→ (xg, g−1y) for all (x, y, g) ∈ F (U )×T (U )×G(U ). Then we define the twisted sheaf F ⊗GT := (F × T )/G.

Example 1.18. For any O∗_X-torsor the twist construction applied to OX gives us a line

bundle. Similarly, twisting O_X⊕nwith a GLn,X-torsor gives a vector bundle.

Using torsors we can define a cohomology theory.

Definition 1.19 ([Gir71, Ch.3 Def. 2.4.2]). Let C be a site and G ∈ Grp(C) a sheaf of groups. We define the non-abelian sheaf cohomology as;

(i) H0(C, G) = Γ(C, G),

(ii) H1(C, G) = {isomorphism classes of left G-torsors on C}.

Remark. Note that H0(C, G) is a group, but H1(C, G) is only a pointed set, where the trivial G-torsor is the distinguished point.

Example 1.20. Denote VBn(X) := {isomorphism classes of vector bundles of rank n}.

Combining examples 1.16 and 1.18 we get morphisms

VBn(X) → H1(X, GLn,X), V 7→ Isom(V, O_X⊕n),

H1(X, GLn,X) → VBn(X), T 7→ O⊕nX ⊗GLn,X T .

These morphisms are inverse constructions, so H1(X, GLn,X) corresponds to the vector

bundles of rank n. Note that in this correspondence we use the same gluing data to realise two locally trivial objects, GLn,S-torsors or vector bundles. This is possible since

automorphisms of the trivial bundle are the invertible matrices.

Theorem 1.21 ([Stacks, Tag 03AG]). When G ∈ Grp(C) is abelian, the cohomology theory defined above coincides with abelian sheaf cohomology.

This justifies our notation H1(C, G) and shows that this cohomology theory extends abelian sheaf cohomology.

Lemma 1.22 ([Gir71, Ch.3 Prop. 1.3.6]). The assignment of H1 is functorial. For G1, G2 ∈ Grp(C) a morphism f : G1 → G2 defines a right action on G1 and induces a

(12)

Short exact sequences induce long exact sequences. In particular when the first group is abelian, we can extend this sequence using abelian sheaf cohomology.

Theorem 1.23 ([Gir71, Ch.4 Cor. 4.3.4]). Let G1, G2, G3 ∈ Grp(C), with G1 abelian. A

short exact sequence

1 → G1 → G2→ G3 → 1

such that the image of G1 is contained in the center of G2 induces a long exact sequence

in cohomology

1 H0(C, G1) H0(C, G2) H0(C, G3)

H1(C, G1) H1(C, G2) H1(C, G3) H2(C, G1).

Note that the H0 _{part is exact as groups, but the rest only as pointed sets.}

The induced long exact sequences in cohomology gives a lot of information, allowing calculations which would otherwise be impossible.

1.3 The Kummer sequence and the projective linear group

There are a few short exact sequences of sheaves in the ´etale topology which which are frequently needed in the thesis. The first sequence of interest is the Kummer sequence, which shows that in the ´etale topology we have the possibility to look at nth-roots of invertible elements, unlike the Zariski topology.

Lemma 1.24 ([Stacks, Tag 03PK]). Let S be a scheme and n ∈ Z>0 such that n ∈ O∗_S,

then the sequence

1 → µn,S → Gm,S ( )n

−−→ Gm,S → 1

is exact in the ´etale site of S. This sequence is called the Kummer sequence.

Sketch of proof. The non-trivial part is showing surjectivity of the last map. This can be done using that n and the elements of Gm,S are invertible, so adding an nth root of

an element of Gm,S is an ´etale morphism.

Remark. Since all schemes in the Kummer sequence take values in abelian groups, this sequence induces an infinite long exact sequence in abelian sheaf cohomology.

Note that by [Stacks, Tag 03P8] we have that Pic(S) = H1(S´et, Gm,S), so the long

exact sequence gives us the following exact sequences

1 → O∗_S(S) ⊗ Z/nZ →H1(S´et, µn,S) → Pic(S)[n] → 1,

1 → Pic(S) ⊗ Z/nZ →H2(S´et, µn,S) → H2(S´et, Gm,S)[n] → 1,

which will be important when we look at the existence and explicit examples of Brauer-Severi varieties.

(13)

Next up, we need the defining sequence of PGLn,S, since the first cohomology group

corresponds to the isomorphism classes of Brauer-Severi schemes, see proposition 2.7. Definition 1.25. Let S be scheme. We define the sheaf PGLn,S ∈ Grp(S´et) by the

short exact sequence of sheaves in the ´etale topology

1 → Gm,S → GLn,S→ PGLn,S → 1.

Remark. This short exact sequence induces a long exact sequence by theorem 1.23. For us, it is relevant to take a closer look at the induced map p∗: H1(S´et, GLn,S) →

H1(S´et, PGLn,S):

We have seen that set H1(S´et, GLn,S) can be interpreted as the isomorphism classes

of vector bundles of rank n over S in the ´etale topology. Since a vector bundle is locally trivial in the ´etale topology if and only if it is locally trivial in Zariski topology, see [Stacks, Tag 05VG], it actually gives isomorphism classes of Zariski vector bundles.

On the other hand, for a scheme U the automorhism group of Pn−1U is given by

PGLn(OU(U )). Therefore we can construct a correspondence of PGLn,S-torsors with

´etale locally trivial projective bundles of rank n − 1 using the same gluing data.

In this framework p∗ is the map sending a vector bundle to its projectivization.

So, if we want to determine whether there exist any projective bundles which are not the projectivization of a vector bundle, we can just check whether the boundary map δ : H1(S´et, PGLn,S) → H2(S´et, Gm,S) is nonzero.

The last sequence we will need is also a sequence involving PGLn,S, which provides

us with more information about the cohomology of PGLn,S.

Lemma 1.26. For a scheme S and n ∈ O∗_S, then sequence 1 → µn,S → SLn,S → PGLn,S → 1

is exact in the ´etale topology. Here the sheaf SLn,S is the representable functor U 7→

SLn(OU(U )).

Sketch of proof. Again only surjectivity of the last map is non-trivial. Using the same argument as for the Kummer sequence, we can add a nth root of unity, which allows us to lift an element from PGLn,S to SLn,S locally.

In section 2.3 we will see how we can combine the two sequences involving PGLn,S to

(14)

2 Existence of non-linear Brauer-Severi

schemes over a surface

The goal of this chapter is to show there exist smooth projective surfaces with non-linear Brauer-Severi schemes. In particular we are interested in finding a criterion for the existence of non-linear Brauer-Severi schemes of a given rank over these surfaces.

First, we will introduce Brauer-Severi schemes and their connection with Azumaya algebras. Then, we will state de Jong’s theorem about the period-index problem and use this to prove that there exist non-linear Brauer-Severi schemes of rank n − 1 over a surface if and only if the Brauer group has non-trivial n-torsion.

2.1 Brauer-Severi schemes and Azumaya algebras

Recall that Brauer-Severi varieties are varieties over a field k which become isomorphic to projective space after a base change to some finite separable field extension K|k. Alternatively, we can define a Brauer-Severi variety as a variety over k which is ´etale locally isomorphic to projective space, since the ´etale topology over a field is given by finite separated field extensions, see example 1.5. This easily generalises to Brauer-Severi schemes.

Definition 2.1 ([Gro95, rk. 8.1]). Let S be a scheme. A Brauer-Severi scheme over S of rank n is a scheme (X → S) ∈ (Sch/S) which is étale locally a trivial projective bundle, i.e. a scheme X étale over S for which there exists an étale cover {Ui → S} such

that for all i the following diagram commutes

PnUi X ×SUi X

Ui S

∼

Remark. This construction gives us a bundle of Brauer-Severi varieties. To see this, take an element of the ´etale cover Ui → X. By definition we have the commutative diagram

as above. Now take s ∈ S and u ∈ Ui such that u 7→ s. Restricting to the fibre above s

gives us

Pnu Xu Xs

Spec(k(u)) Spec(k(s))

(15)

Note that the bottom map is ´etale and hence standard ´etale. Therefore, k(u) = k(s)[X]/(f ) where f is separable and irreducible, since k(u) is a field. Hence the map Xu → Xs is given by a base change to a finite separable field extension of k(s) and

over this extension Xu ∼= Pnu, thus Xs is a Brauer-Severi variety. In particular, over

S = Spec(k) we recover the classical definition of a Brauer-Severi variety.

Example 2.2. One way of obtaining Brauer-Severi schemes is by taking the projec-tivization of a vector bundle, the linear Brauer-Severi schemes.

The simplest non-linear Brauer-Severi variety is a smooth conic over k without k-points, such as C = {x2 + y2+ z2 = 0} ⊂ P2_R. Note that a conic containing a point can be parametized, which gives an isomorphism with the projective line. Thus, a conic without points becomes isomorphic to the projective line over a finite field extension which adds a point on the conic. Hence, it is a Brauer-Severi variety.

We can generalise this to smooth conic bundles, see sections 3.1 and 3.2.

Like the bundles of Brauer-Severi varieties, we can define bundles of central simple algebras. Recall that as a result of Wedderburn’s theorem, an algebra A over a field k is central simple if and only if there exists an integer n > 0 and a finite field extension K|k such that A ⊗kK ∼= Mn(K) [Stacks, Tag 074X]. Based on this fact, we define a

bundle of central simple algebras as follows:

Definition 2.3 ([Stacks, Tag 0A2J]). Let S be a scheme. An OS-algebra A is called

Azumaya if it is ´etale locally a matrix algebra, i.e. when there exists an ´etale cover {ϕi: Ui → S} such that ϕ∗_iA ∼= Mdi(OUi) for some di > 0.

Proposition 2.4 ([Gro95, Thm. 5.1]). Let A be an OS-algebra of finite type as OS

-module, then the following are equivalent; (i) A is an Azumaya OS-algebra, and

(ii) A is locally free as OS-module and for all x ∈ S the fibre A(x) = Ax⊗OS,xk(x) is

a central simple algebra over k(x).

Remark. This gives that an Azumaya OS-algebra is a bundle of central simple algebras.

Example 2.5 (see e.g. [GS17]). Let k be a field, then quaternion algebras are examples of central simple algebras and therefore Azumaya algebras over Spec(k).

For two elements a, b ∈ k∗ define the (generalised) quaternion algebra (a, b)k as the

4-dimensional k-algebra with basis 1, i, j, ij and relations i2 = a, j2 = b, ij = −ji.

Note that the isomorphism class of the algebra (a, b)k only depends on the classes of a, b

in k∗/(k∗)2.

Moreover, we have an isomorphism from (1, b)k to M2(k) given by

i 7→1 0 0 −1 , j 7→0 b 1 0 .

(16)

We can conclude that (a, b)k⊗ k[

√

b] ∼= M2(k[

√

b]). This can be generalised to bundles, see section 3.2.

Definition 2.6. Let S be a connected scheme. The degree of an Azumaya OS-algebra A

is the natural number n such that A is locally isomorphic to Mn(OS), denoted deg(A).

There is a one-to-one correspondence between isomorphism classes of Brauer-Severi schemes of rank n − 1 and Azumaya algebras of degree n. Denote Azn(S) for the set of

isomorphism classes of degree n Azumaya OS-algebras and BSn−1 for the isomorphism

classes of Brauer-Severi schemes of rank n − 1.

Proposition 2.7 ([Gro95, rk. 8.1]). Let S be a connected scheme, then Azn(S) ∼= H1(S´et, PGLn,S) ∼= BSn−1(S),

as pointed sets.

Sketch of proof. By Skolem-Noether Aut(Mn) = PGLn ([Stacks, Tag 074P]), therefore

we can use the Isom and twist construction on Mn(OS) to get bijection H1(S´et, PGLn,S) ∼=

Azn(S).

On the other hand, since also Aut(Pn−1) = PGLn we can construct an isomorphism

H1(S´et, PGLn,S) ∼= BSn−1(S) using the gluing data from the torsors to glue projective

bundles and vice versa.

Remark. Note that over S = Spec(k), we recover the correspondence between Brauer-Severi varieties and central simple algebras over k.

Using this correspondence between Brauer-Severi schemes and Azumaya algebras we can study them interchangeably. In the next sections we will recollect some result about Azumaya algebras, which will give us the information about Brauer-Severi schemes we need prove which surfaces admit non-linear Brauer-Severi schemes of rank n.

2.2 The Brauer group and the period-index problem

An important notion in the study of central simple algebras is the Brauer group of a field k. The Brauer group is the set of Morita equivalent central simple algebras, which form a group under tensor product. A natural question about elements of the Brauer group is the period-index problem, which remains to be an open problem in many cases. In this section, we will introduce the objects and definitions needed to state the period-index problem over a field, the relevant results and how we can apply this to Azumaya algebras. Moreover, we will relate the Brauer group of a scheme to the cohomological Brauer group, to see that the trivial Brauer class corresponds with the projectivization of vector bundles.

Definition 2.8 (See e.g. [GS17]). Let A, A0 be central simple algebras over a field k. (i) By Wedderburn’s theorem there exists a division algebra D such that A ∼= Mn(D)

(17)

(ii) Two central simple k algebras A, A0 are called Brauer equivalent if A ⊗kMn(k) ∼=

A0⊗kMn0(k) for some n, n0 ∈ Z_>0. Note that this happens if and only if D ∼= D0.

(iii) We can define addition on the equivalence classes of central simple algebras by [A] + [A0] = [A ⊗kA0]. This turns the set of equivalence classes of central simple

algebras into a group, called the Brauer group of a field, denoted Br(k). (iv) The period of A, denoted per(A), is defined as the order of [A] in Br(k).

The period-index problem arises from the following result.

Theorem 2.9 ([GS17, Prop. 4.5.13], Brauer). Let A be central simple algebra over a field k. Then per(A) | ind(A). Moreover, per(A) and ind(A) contain the same prime factors.

This implies that there exist a integer d such that ind(A) | per(A)d. The period-index problem is the question about which d occur and in particular when d equals 1, so that the period equals the index. For a function field of a variety of dimension d, it is conjectured that ind(A) | per(A)d−1[Sta08, Ch.4 1.1]. The conjecture about function fields of varieties is based on the results for curves and surfaces:

Theorem 2.10 ([Stacks, Tag 0A2M], Tsen). For a function field K of a curve over an algebraically closed field we have Br(K) = 1.

Remark. Thus, in the Brauer group of a curve we have ind(A) | per(A)0.

Theorem 2.11 ([Jon04], de Jong). Let k be a separably closed field and let K|k be a finitely generated field extension of transcendence degree 2. Let A a central simple algebra over K with period prime to the characteristic of k. Then per(A) = ind(A). Remark. Later de Jong and Starr proved that the assumption that the period has to be prime to the characteristic is not needed ([JS10, Thm. 1.0.1] or [Sta08, Ch.4] for explanation).

Using Azumaya algebras we can construct the Brauer group of a scheme; for which there also exists a period-index problem. It turns out that this problem is related to the period-index problem for central simple algebras. But first, we need the definitions. Definition 2.12 ([Stacks, Tag 0A2J]). Let S be a connected scheme and A, A0 Azumaya OS-algebras.

(i) The degree of A, denoted deg(A), is the integer n such that A is locally isomorphic to Mn(OU).

(ii) The Azumaya algebras A, A0 are called Brauer equivalent if there exist integers n, n0 such that A ⊗OSMn(OS) ∼= A

0_⊗

OS Mn0(OS).

(iii) Again, we can define addition on the equivalence classes by [A] + [A0] = [A ⊗OSA

0_].

This turns the set of equivalence classes of Azumaya algebras into a group, called the Brauer group of a scheme, denoted Br(S).

(18)

(iv) The period of A, denoted per(A), is defined as the order of [A] in Br(S). (v) The index of A is defined as ind(A) := gcd{deg(A0) | [A] = [A0]}.

Remark. By proposition 2.7 we get Azn(S) = H1(S´et, PGLn,S). We can use this to get a

more direct description of Brauer group: Note that the map GLn,S → GLnm,S sending a

matrix M to the block matrix with m copies of M on the diagonal descends to a map on PGLn,S → PGLnm,S. The induced map H1(S´et, PGLn,S) → H1(S´et, PGLnm,S) is given

by A 7→ A ⊗OSMn(OS). Hence Br(S) = lim_−→H

1_(S ´

et, PGLn,S). Thus the Brauer group is

the direct limit of the first cohomology sets H1(S´et, PGLn,S).

On the other hand, proposition 2.7 gives H1_(S

´et, PGLn,S) = BSn−1(S). This way, the

Brauer group gives us information about the classes of Brauer-Severi schemes over S. Remark. In the case of central simple algebras, the index can also be defined by ind(A) = min{deg(A0) | [A] = [A0]}. This shows how the definition of the index for an Azumaya algebra generalises the index of central simple algebras. The problem is that for Azumaya algebras it is not clear whether there exits an Azumaya algebra equivalent to A of degree equal to the index of A. It turns out that this is not true for schemes of dimension > 2, see [AW14b, Thm.1.1] for a counter example.

Theorem 2.13 ([Jon04, Step 3]). Let S be a smooth projective surface over an alge-braically closed field k, then for any Azumaya OS-algebra A there exists an Azumaya

OS-algebra A0 such that deg(A0) = ind(A) and [A] = [A0] in the Brauer group. In other

words, ind(A) = min{deg(A0) | [A] = [A0]}.

When our base scheme is nice, it turns out that there is an inclusion of the Brauer group over a scheme to the Brauer group of the function field of the scheme. This allows us to connect the notions of period and index of an Azumaya algebra with the ones we have seen for central simple algebras.

Theorem 2.14 ([Mil80, Cor. IV 2.6]). Let S be a regular integral scheme, then the natural map Br(S) ,→ Br(k(S)) given by the restriction to the generic point, is injective.

Remark. Because this map is injective, it preserves the period of an element. In general this map is not surjective. Therefore, it is not clear whether the index is preserved, only that for A(η) we have ind(A(η)) | ind(A), where η is the generic point of S.

Theorem 2.15 ([AW14a, Prop. 6.1]). Let S be a regular noetherian scheme, then ind(A) = ind(A(η)), where η is the generic point of S.

Remark. We can conclude that for connected regular noetherian scheme S the period and index in Br(S) coincide with the period and index of the image in Br(k(S)). This allows us to use results about the period-index problem for function fields.

Besides the Brauer group there also exists the cohomological Brauer group. As their names suggest, the two Brauer groups are related.

(19)

Definition 2.16 ([Stacks, Tag 0A2J]). The cohomolgical Brauer group of a scheme S is defined as Br0(S) := H2(S´et, Gm,S)tors, the torsion subgroup of H2(S´et, Gm,S).

Recall that the short exact sequence 1 → Gm,S → GLn,S → PGLn,S → 1 induces a

long exact sequence in cohomology, as described at 1.25. We get the exact sequence H1(S´et, GLn,S) → H1(S´et, PGLn,S)

δn

−→ H2(S´et, Gm,S),

where the map H1(S´et, GLn,S) → H1(S´et, PGLn,S) can be interpreted as the map sending

a vector bundle to its projectivization. Moreover, using that Br(S) is the direct limit of the sets H1(S´et, PGLn,S), the maps δn induce a map δS: Br(S) → H2(S´et, Gm,S). This

map relates the Brauer group and the cohomological Brauer group.

Theorem 2.17 ([Stacks, Tag 0A2J], ). The map δS is injective. Moreover, when S is

quasi-compact or connected, Br(S) is a torsion group and therefore its image lies in the cohomological Brauer group Br0(S).

Remark. From the definition of δS and the fact that it is injective, we see that a

Brauer-Severi scheme is linear if and only if it defines the trivial class in the Brauer group. For example, if S is a connected regular curve over an algebraically closed field, Tsen’s theorem 2.10 gives that all Brauer-Severi schemes over S are linear.

Based on a result of Gabber, de Jong proved an even stronger result.

Theorem 2.18 ([Jon03]). Let S be a quasi-projective scheme, then Br(S) surjects onto Br0(S), so Br(S) = Br0(S).

Thus, if S is quasi-projective we can use the above result to see that we actually get an exact sequence

H1(S´et, GLn,S) → H1(S´et, PGLn,S) δn

−→ Br(S).

In the next section we will combine this sequence with de Jong’s result on the period-index problem over a surface to show that δn surjects into the n-torsion of the Brauer

group.

2.3 Brauer-Severi schemes over a surface

We have seen that by Tsen’s theorem all Brauer-Severi schemes over a curve are linear. This raises the question whether all Brauer-Severi schemes are linear. We will use de Jong’s theorem to show that this is not true for smooth projective surfaces.

Proposition 2.19. Let S be a smooth projective surface over an algebraically closed field k and n ∈ Z>0 with char(k) - n, then the inclusion δn: H1(S´et, PGLn,S) → Br(S)

surjects onto the n-torsion of Br(S). Moreover, it fits into an exact sequence H1(S´et, GLn,S) → H1(S´et, PGLn,S) → Br(S)[n] → 1.

(20)

Proof. Without loss of generality we can assume that S is connected.

We can combine the exact sequences from 1.26 and 1.25 into the commutative diagram:

1 Gm,S GLn,S PGLn,S 1

1 µn,S SLn,S PGLn,S 1,

where the vertical arrows are given by the inclusion maps. This induces long exact sequences in cohomology. Since S is quasi-projective we can use theorem 2.18, to get the commutative diagram

H1(S´et, GLn,S) H1(S´et, PGLn,S) Br(S)

H1(S´et, SLn,S) H1(Set´, PGLn,S) H2(S´et, µn,S). δn

This implies that the map δn factors through H1(S´et, µn,S). Since all elements of µn,S

have an order dividing n, the elements of H1_(S ´

et, µn,S) are n-torsion and the image of

δn lies inside Br(S)[n]. Hence we get the exact sequence

H1(S´et, GLn,S) → H1(S´et, PGLn,S) δn

−→ Br(S)[n]. It is left to show that δn is surjective.

Because S is a connected regular noetherian scheme we can combine theorems 2.14 and 2.15 to see that the map Br(S) ,→ Br(k(S)) preserves the period and index. Moreover, we can combine this with de Jong’s theorem (2.11) on Br(k(S)) to see that the period equals the index in Br(S).

Suppose we have an [A] ∈ Br(S)[n], then per([A]) = m | n. By de Jong’s theorem also ind([A]) = m and theorem 2.13 implies that we have a A0 ∈ H1_(S

´et, PGLm) with

[A] = [A0]. Now A0⊗M_n/m(OS) ∈ H1(Set´, PGLn,S) satisfies [A0⊗M_mn(OS)] = [A0] = [A],

by definition of the Brauer equivalence, which shows that δn is surjective.

Remark. The proposition implies that there exist non-linear Brauer-Severi schemes of rank n − 1 over S if and only if the Brauer group has non-trivial n-torsion points. In particular, it gives that there exist at least #Br(S)[n] − 1 isomorphism classes of non-linear Brauer-Severi schemes of rank n − 1.

Examples of smooth projective surfaces with non-trivial n-torsion in the Brauer group for all n ∈ Z>0 are K3 surfaces. This gives that the there exist surfaces with non-linear

Brauer-Severi schemes of any rank over these surfaces.

A natural follow-up question is whether we can find explicit examples of non-linear Brauer-Severi schemes, in particular over a surface. Moreover, to which class in the Brauer group does this scheme correspond? Chapter 3 is all about the answer to these questions, especially about finding explicit constructions of non-linear Brauer-Severi schemes.

(21)

3 Examples of Brauer-Severi schemes

The obvious way of obtaining Brauer-Severi schemes is by the projectivization of a vector bundle. In this section we look at examples of Brauer-Severi schemes not of this form, the non-linear Brauer-Severi schemes. We start with an introductory example, the universal diagonal conic. Then, construct conic bundles from Azumaya algebras of degree two, created out of double covers. Moreover, we give the corresponding class in the Brauer group explicitly. We conclude with giving a general construction from short exact sequences of finite groups contained in 1 → Gm,S → GLn,S → PGLn,S → 1, which

give Brauer-Severi schemes of any rank.

3.1 The universal diagonal conic

The first example of a non-linear Brauer-Severi variety over a field is the conic without points. One way of generalising this, is by looking at conic bundles. A standard example of a conic bundle is the universal conic. Inspired on this we will look at the universal diagonal conic, which is defined over a surface. Note that all conics over a field of characteristic 2 are reducible, therefore we will not work over a field of characteristic 2. Example 3.1. Let k be a field of characteristic unequal to two. We define the universal diagonal conic C to be the subscheme of P2k× P2kcut out by the ideal (aX2+ bY2+ cZ2),

where a, b, c are the coordinates of the first factor and X, Y, Z the coordinates of the second factor. Note that C lies over P2k by projection onto the first factor.

Since our aim is to build a Brauer-Severi scheme we are only interested in smooth conics. Note that a conic aX2+ bY2+ cZ2 is smooth if and only if it is irreducible, which is if all coefficients a, b, c are nonzero. Thus, we are interested in the part Csm:= CV → V

where V = Proj(k[a, b, c]abc) ∼= Spec(k[α±1, β±1]), here the isomorphism is induced by

α 7→ a/c and β 7→ b/c. Denote A := k[α±1, β±1], then we get Csm= Proj(A[X, Y, Z]/(αX2+ βY2+ Z2)).

We will show that the scheme Csm→ V defines a non-linear Brauer-Severi scheme.

Lemma 3.2. The scheme Csm → V defines a Brauer-Severi scheme of rank 1.

Proof. Let U = Spec(B), where B = A[γ]/(γ2 + α), and consider the map U → V induced by the integral extension A → B. This map is standard ´etale since γ2+ α is monic and 2γ is invertible since both 2, α ∈ A∗. Since an integral exension induces a surjective map on spectra we get that U → V defines an ´etale cover.

(22)

Now (Csm)U ∼= Proj(B[X, Y, Z]/(−X2 + βY2 + Z2)), using that γ2 = −α and γ is

invertible to absorb the factor −α into X2. This allows us to parametrize the conic −X2_{+ βY}2_{+ Z}2 _{using the graded ring map}

ϕ : B[X, Y, Z]/(−X2+ βY2+ Z2) → B[u, v], generated by: X 7→ u2+ βv2, Y 7→ 2uv Z 7→ u2− βv2.

Since 2 and β are invertible this morphism induces an isomorphism of the degree n part of B[X, Y, Z]/(−X2+ βY2+ Z2) with the degree 2n part of the graded ring B[u, v] therefore, the map ϕ induces an isomorphism on Proj. Hence (Csm)U ∼= P1U, which shows

that Csm is a Brauer-Severi scheme of rank 1.

Lemma 3.3. The Brauer-Severi scheme Csm→ V is not the projectivization of a vector

bundle.

Proof. It suffices to show that Csm has no rational sections, so let us look above the

generic point of η of V . So we need to look at the k(α, β)-rational points of (Csm)η = Proj(k(α, β)[X, Y, Z]/(αX2+ βY2+ Z2)).

But these do not exist, since only solution of αX2+ βY2+ Z2 = 0 in k(α, β) is the trivial one.

Remark. We can conclude that Csm is indeed a non-linear Brauer-Severi scheme of rank

1. Moreover, using that Pic(V ) = 0 and V has the cohomology type of a torus one can show that Br(V )[2] has only one element, so Csm → V represents the only non-trivial

Brauer class. In the next section we will give a method to find the class in the Brauer group directly.

3.2 Constructing Brauer-Severi schemes from double

covers

Inspired on the construction of quaternion algebras and their associated conic, we will construct an Azumaya algebra of degree 2 from two elements of H1_(S

´

et, µ2,S) and a

corresponding Brauer-Severi scheme of rank 1. Furthermore, we show that the Brauer class is given by the cup product. Moreover, we introduce a more symmetric construction of Brauer-Severi schemes from three double covers and show they are isomorphic to Brauer-Severi schemes obtained from two covers. Throughout this section we require 2 to be invertible on our base scheme S.

3.2.1 A construction from two double covers

Let S be a scheme. We will construct an Azumaya algebra of degree 2 from two elements of H1(S´et, µ2,S). It is based on the assignment of the quaternion algebra (a, b)k (from

example 2.5) to a, b ∈ k∗/(k∗)2 ∼= H1(S´et, µ2,S).

(23)

Lemma 3.4 ([Stacks, Tag 040Q]). Let S be a scheme where 2 ∈ O∗_S. There is a canonical identification of H1(S´et, µ2,S) with the group of pairs (L, α) where L ∈ Pic(S)

and α : L⊗2 ∼−→ O_S.

Remark. Using this description it is clear how H1(S´et, µ2,S) fits into the sequence

1 → OS(S)∗/(OS(S)∗)2→ H1(S´et, µ2,S) → Pic(S)[2] → 1,

induced by the Kummer sequence.

Moreover, we can use the identification α : L⊗2 −→ O∼ S to turn OS ⊕ L into a OS

-algebra. Now T = Spec_S(OS⊕ L) defines a double cover of S. So, one can interpret the

elements µ2,S-torsors as double covers.

Definition 3.5. Let (L, α), (M, β) ∈ H1(S´et, µ2), then we define the OS-algebra (α, β)S

as the OS-module OS⊕L⊕M⊕(L⊗M), where we define the multiplication by sending a

pair of elements to their tensor product, using the identifications l1⊗l2= α(l1⊗l2) ∈ OS,

m1⊗ m2= β(m1⊗ m2) ∈ OS and m ⊗ l = −l ⊗ m ∈ L ⊗ M.

Remark. The product is well defined and defines an OS-algebra structure since the tensor

product is associative and distributive, α, β are OS-module morphisms and 1 ∈ OS

defines the unit element.

Note that L and M are implicit in the notation (α, β)S, since α, β are morphisms

from L resp. M to OS.

Lemma 3.6. The OS-algebra (α, β)S is an Azumaya OS-algebra.

Proof. Let x ∈ S. By construction A(x) is a quaternion algebra over k(x), hence A is an Azumaya OS-algebra by proposition 2.4.

Remark. Over Spec(k) the line bundles are trivial and the trivializations α, β correspond to elements a, b ∈ k∗/(k∗)2. Now the Azumaya algebra (α, β)Spec(k) is by definition the

quaternion algebra (a, b)k.

The conic associated to the quaternion algebra (a, b)k is given by {ax2+ by2− abz2} ⊂

P2k, where P2k = P((a, b)tr=0k ). Inspired on the construction we associate a conic bundle

to (α, β)S.

Definition 3.7. Let (L, α), (M, β) ∈ H1(S´et, µ2). Define BS(α, β) as the closed

sub-scheme of P(Lx ⊕ My ⊕ (L ⊗ M)z) cut out by the ideal sheaf (αx2+ βy2− αβz2_).

Lemma 3.8. The scheme BS(α, β) is a Brauer-Severi scheme of rank 1.

Proof. This is an ´etale local property, so without loss of generality we can assume that S is affine and (L, α) and (M, β) are trivial. Now the proof reduces to a simpler version of the proof for the universal diagonal conic, see lemma 3.2.

Lemma 3.9. Let (L, α), (M, β) ∈ H1(S´et, µ2). The Azumaya algebra (α, β)S and the

Brauer-Severi scheme BS(α, β) define the same class in H1_(S ´

(24)

Proof. It suffices to show that they can be constructed from the same gluing data. Let us fix a cover {Up → S} trivialising the pairs (L, α), (M, β), with corresponding gluing

data (spq, Upq), (tpq, Upq) ∈ ˇH1(S´et, µ2,S).

Let us start with constructing the gluing data for (α, β)S. For this, note that for any

scheme X we have an identification (1, 1)X = OX⊕ OXi ⊕ OXj ⊕ OXij ∼= M2(OX) by

sending i 7→1 0 0 −1 , j 7→0 1 1 0 .

Applying this identification gives us that the gluing maps of (α, β)S are given by

gpq: 1 0 0 −1 7→ s_pq1 0 0 −1 , gpq: 0 1 1 0 7→ t_pq0 1 1 0 .

Using that spq, tpq are maps from Upq to µ2,S we get for each x ∈ Upq the gluing maps

gpq acts by conjugating with the matrix:

spq(x), tpq(x) 1, 1 −1, 1 1, −1 −1, −1 gpq(x) 1 0 0 1 1 0 0 −1 0 1 1 0 0 1 −1 0

On the other hand, using how we defined BS(α, β) we get that the gluing map on the trivialisation OUpq[X, Y, Z]/(X

2_{+ Y}2_{− Z}2_{) is given by X 7→ s}

pqX, Y 7→ tpqY, Z 7→

spqtpqZ. Since the automorphisms of projective space are defined up to scaling, we can

multiply by spqtpq and get X 7→ tpqX, Y 7→ spqY, Z 7→ Z. Combining this with the

identification to projective space from 3.2 given by ϕ : OUpq[X, Y, Z]/(X

2_{+ Y}2_{− Z}2_{) → O}

Upq[U, V ],

generated by: X 7→ U2− V2, Y 7→ 2U V Z 7→ U2+ V2.

we see that the gluing map is induced by acting with the matrix gpq on OUpq[U, V ].

Remark. We can conclude that indeed did construct a Brauer-Severi scheme BS(α, β) corresponding to the Azumaya algebra (α, β)S. Moreover, note that over Spec(k) this

correspondence reduces to associating a conic to a quaternion algebra.

3.2.2 A construction from three double covers

A more symmetric way of the previous construction is one where we use three covers instead of two.

Definition 3.10. Let (L, α), (M, β), (N , γ) ∈ H1_(S ´

et, µ2,S) and define BS(α, β, γ) ⊂

P(Lx ⊕ My ⊕ N z) be the closed subscheme cut out by the ideal sheaf (αx2+ βy2+ γz2). Lemma 3.11. The scheme BS(α, β, γ) is a Brauer-Severi of rank 1 over S.

(25)

This construction is more general than the construction from two covers, but it turns out they define the same torsors because each Brauer-Severi scheme constructed from three covers is isomorphic to one constructed from two double covers. Intuitively this is because the conic {ax2+by2−abz2 _{= 0} ⊂ P}2 _{is isomorphic to {ax}2_+by2_−z2 _{= 0} ⊂ P}2_,

while on the other hand {ax2+by2+cz2 = 0} ⊂ P2is isomorphic to {−acx2−bcy2_−z2₌

0} ⊂ P2.

Lemma 3.12. Let S be noetherian scheme and (L, α), (M, β), (N , γ) ∈ H1(S´et, µ2,S).

There exists an isomorphism between BS(α, β, γ) and BS(−αγ, −βγ).

Proof. Recall that for a locally free sheaf E on S and L ∈ Pic(S) we have a natural isomorphism P(E) ∼= P(E ⊗ L) induced by the local trivialisations of L [Har77, Ch.II Lemma 7.9] (note that this still holds for our convention: P(E) = Proj(Sym•(E∨))).

Applying this to Lx ⊕ My ⊕ N z and the line bundle N induces an isomorphism P(Lx⊕My⊕N z)−→ P(L⊗N x⊕M⊗N y⊕N∼ ⊗2z) sending BS(α, β, γ) to BS(γα, γβ, γ2). Moreover, using γ : N⊗2 ∼−→ O_S _{we get an isomorphism P(L⊗N x⊕M⊗N y ⊕N}⊗2_z)₋∼_→

P(L ⊗ N x ⊕ M ⊗ N y ⊕ OSz) sending BS(γα, γβ, γ2) ∼

−→ BS(γα, γβ, 1), where 1 : O_S⊗2−∼→ O_S the standard morphism given by multiplication. Hence BS(α, β, γ) ∼= BS(γα, γβ, 1). On the other hand, applying the same lemma on Lx⊕My ⊕(L⊗M)z with L⊗M and applying the trivialisations α and γ gives us an isomorphism BS(α, β)−∼→ BS(β, α, −1). Note that BS(α, β) ∼= BS(β, α) by construction. Combining these findings gives us an isomorphism BS(α, β, γ) ∼= BS(−αγ, −βγ).

Remark. In the first step of the proof we made the choice to use the line bundle N , but we could have used any of the line bundles. This gives us isomorphisms

BS(α, β, γ) ∼= BS(−αγ, −βγ) ∼= BS(−βα, −γα) ∼= BS(−γβ, −αβ).

3.2.3 The class in the Brauer group

Now that we can construct Brauer-Severi schemes, a natural question is whether they are non-linear, or specifically which class they define in the Brauer group. By lemmas 3.9 and 3.12 we only need to consider the class of (α, β)S to determine the class in the

Brauer group for both constructions.

Proposition 3.13. Let S be a scheme such that 2 ∈ O∗_S and take (L, α), (M, β) ∈ H1(S´et, µ2,S), then (α, β)S = α ∪ β ∈ H2(S´et, Gm,S).

Remark. Over S = Spec(k) the proposition gives that the class of the quaternion alge-bra (a, b)k is given by a ∪ b ∈ H1(Spec(k)´et, µ2,S). This was already know, see [GS17,

Prop. 4.7.1], but this proof does not generalise to the relative case since it is asymmetric in a, b. The proof proposition 3.13 gives an alternative proof of this fact, one which is symmetric in a and b.

Proof. Note that we always have a morphism from ˇCech cohomology to ´etale cohomology moreover, in case of the first cohomology group this is a isomorphism [Stacks, Tag 03OU]. Hence it suffices to show the identity in ˇCech cohomology.

(26)

Since 2 ∈ O∗_S we have a global primitive 2nd rood of unity, −1 ∈ OS, which gives us

an identification µ2,S ∼= Z/2Z. We will use to interchange the coefficients µ2,S to Z/2Z.

Let us fix a cover {Ui → S} trivialising the pairs (L, α), (M, β), with corresponding

gluing data (sij, Uij), (tij, Uij) ∈ ˇH1(S´et, Z/2Z). The cup product is given by α ∪ β =

(sij, Uij) ∪ (tij, Uij) = (sij ⊗ tjk, Uijk) ∈ ˇH2(S´et, Z/2Z ⊗ Z/2Z). Now applying the

identification Z/2Z ⊗ Z/2Z −_{→ Z/2Z given by a ⊗ b 7→ ab to the cup product gives}∼ α ∪ β = (sijtjk, Uijk) ∈ ˇH2(S´et, Z/2Z). Using the identification of µ2,S with Z/2Z

and the inclusion of µ2,S into Gm,S the cup product α ∪ β corresponds to the element

((−1)sijtjk_{, U}

ijk) ∈ ˇH2(S´et, Gm,S).

We will show that the Azumaya algebra (α, β)Sdefines the same class in ˇH2(S´et, Gm,S).

Recall that for an Azumaya algebra A ∈ H1(S´et, PGLn,S) we get its class in H2(S´et, Gm,S)

by computing the image of the boundary map coming from the short exact sequence 1 → Gm → GLn,S → PGLn,S → 1.

By the proof of lemma 3.9 we get that the gluing data of (α, β)S is given by

gij = 1 0 0 −1 tij_{0 1} 1 0 sij ∈ PGL₂(Uij).

To compute the image of the boundary map we need to lift (gij, Uij) to GL2,S. Let us

choose the most obvious lift, given by ˜gij =

1 0 0 −1 tij 0 1 1 0 sij ∈ GL_n,S(Uij).

It turns out that we can simplify the situation by noting that h1 0 0 −1 ,0 1 1 0 i is isomorphic to D4 = hσ, ρ | σ2 = ρ4 = (σρ)2 = ei where the isomorphism is given by

1 0 0 −1 7→ σ, 0 1 1 0 7→ σρ. Using this, we can write ˜gij = σtij(σρ)sij.

The next step is to calculate (˜gij˜gjk˜g−1ik , Uijk). Note that since (sij, Uij), (tij, Uij) form

gluing data we have sik = sij + sjk and tik = tij + tjk on Uijk. Using this in the first

step and then the relation ρσ = σρ−1 to move σtjk _{to the left we get}

˜

gijg˜jk˜g−1_ik = σtij(σρ)sijσtjk(ρ−1σ)sijσtij+tjk

= σsij+tij+tjk_ρsij_(ρ2₎sijtjk_(ρ−1₎sij_σsij+tij+tjk _{= (ρ}2₎sijtjk

This lifts to the cycle ((−1)sijtjk_{, U}

ijk) ∈ ˇH2(S´et, Gm,S). Hence we can conclude that

(α, β)S= ((−1)sijtjk, Uijk) = α ∪ β ∈ ˇH2(S´et, Gm,S), just as desired.

Remark. Taking a more careful look at the proof of 3.13 reveals that our Azumaya algebra (α, β)Sactually comes form an element in H1(S´et, µ2,S×µ2,S) under the inclusion

µ2,S× µ2,S,→ PGL2,S given by (s, t) 7→ 1 0 0 −1 t 0 1 1 0 s

(27)

µ2,S×µ2,S-cocycle to D4and finally land in µ2. This reveals the underlying commutative

diagram

1 µ2,S D4 µ2,S× µ2,S 1

1 Gm,S GL2,S PGL2,S 1.

Hence, this construction arises from the above short exact sequence of finite groups, which is contained in 1 → Gm,S → GLn,S → PGLn,S → 1. The essential observation is

that this sequence is non-split. In a split sequence the boundary map is trivial, since the lift of a cocycle is again a cocycle, therefore it only gives linear Brauer-Severi schemes.

Moreover, using that H1(S´et, µ2,S)2 = H1(S´et, µ2,S× µ2,S), the proof gives that the

boundary map δ : H1(S´et, µ2,S)2 → H2(S´et, µ2,S) coincides with the cup product on

H1_(S

´et, µ2,S).

When S is quasi-compact or connected we can use theorem 2.17, to see that proposition 3.13 gives us a simple way to check whether the Brauer-Severi schemes BS(α, β, γ) are non-linear and which class they define in the Brauer group, namely −γα ∪ −γβ ∈ Br0(S) ⊂ H2_(S

´et, Gm,S). In particular, since the cup product is in general non-trivial,

our constructions do give non-linear Brauer-Severi schemes over certain base schemes.

3.3 Constructions arising from short exact sequence of

finite groups

In the previous section we constructed Azumaya algebras and Brauer-Severi schemes from double covers. From the proof of proposition 3.13 we could deduce that this con-struction arose from a non-split short exact sequence of finite groups. This can be generalised by looking at arbitrary short exact sequences of finite groups contained in 1 → Gm,S → GLn,S → PGLn,S → 1. We will focus on the sequences of constant

fi-nite groups, because they allow a nice explicit description of the induced Brauer-Severi scheme.

Remark. Let S be a scheme, and GSa constant finite group scheme contained in PGLn,S.

Then we can give an explicit description of the map H1(S´et, GS) → H1(S´et, PGLn,S):

Since GS is constant and finite it is affine (combine [Stacks, Tag 03YW], [Stacks, Tag

03P5]). Then, using faithfully flat descent for modules [Stacks, Tag 03O6] gives that each GS-torsor is representable by a GS-cover T . Moreover, since the diagonal action

of GS on T ×SPn−1_S is free we can use [Stacks, Tag 07S7] to see that (T ×SPn−1_S )/GS

is a scheme. Note that, by construction (T ×S Pn−1_S )/GS has the same gluing data

as T ⊗GS PGLn,S and forms a Brauer-Severi scheme of rank n − 1. Hence, this is the

Brauer-Severi scheme which is the image of T under H1(S´et, GS) → H1(S´et, PGLn,S).

Construction 3.14. Let S be a scheme and HS and GS constant finite group schemes

(28)

that we get the following commutative diagram with exact rows:

1 µm,S HS GS 1

1 Gm,S GLn,S PGLn,S 1.

Each such diagram induces a construction of Brauer-Severi schemes by sending a GS

-cover T to (T ×SPn−1_S )/GS.

If the sequence 1 → µm,S → HS → GS → 1 is split all induced Brauer-Severi schemes

will be linear, hence we want the sequence to be non-split. Moreover, suppose m ∈ O_S∗, then the Kummer sequence gives that a torsor T ∈ H1(S´et, GS) induces a non-linear

Brauer-Severi scheme if and only if its image under δG: H1(S´et, GS) → H2(S´et, µm,S) is

not contained in the image of Pic(S) ⊗ Z/mZ in H2(S´et, µm,S).

Using such short exact sequences one can give an explicit construction of Brauer-Severi schemes of any rank. Let’s give an illustration.

Example 3.15. Let S be a scheme such that 2 ∈ O∗_S. An example of a sequence of finite groups as in 3.14 given by

1 µ2,S D2n Dn 1 1 Gm,S GLn,S PGLn,S 1, where D2n= hρ =        0 1 0 · · · 0 0 0 1 · · · 0 .. . ... . .. . .. ... 0 0 · · · 0 1 −1 0 · · · 0 0        , σ =       0 · · · 0 1 .. . . .. 0 0 . .. ... 1 0 · · · 0       i ⊂ GLn,S

and Dn is the reduction of this subgroup in PGLn,S. This sequence is clearly non-split,

and therefore gives us a construction of possibly non-linear Brauer-Severi schemes of rank n − 1. Explicitly, a Dn-covers T gets send to (T ×SPn−1_S )/Dn.

Thus, using this framework we can find Brauer-Severi schemes of any rank with an elegant explicate description, given by a quotient. Also, it simplifies the computation to determine whether the obtained Brauer-Severi scheme is non-linear.

There are still many more examples to explore. Furthermore, this construction can easily be generalised, e.g. allow non-constant or infinite subgroups. We chose to focus on this example, as it was best suited for our purpose, to giving explicit constructions of non-linear Brauer-Severi schemes. Moreover, this construction summarises what we have done: the universal diagonal conic is an example of the construction from double covers, and the construction from double covers is a special case of the construction from short exact sequences of finite groups. Additionally, we can conclude that the Brauer-Severi schemes BS(α, β, γ) are isomorphic to quotients of V4-covers.

(29)

Popular summary

In algebraic geometry one often prefers to work in projective space because it has nice properties. Projective space P(V ) can be viewed as the set of lines through the origin of a vector space V . We are interested in spaces which are almost a projective space: they become isomorphic to projective space once you extend your scalars, i.e. replace real numbers by complex numbers. These objects are called Brauer-Severi varieties.

An example is the rational circle {(x, y) ∈ Q2| x2_+y2 _{= 1} in R}2_{. We can parametrize}

all the rational points on the unit circle by looking at the second intersection of the unit circle with a line with rational slope through (−1, 0), counted with multiplicity.

Since the elements of P(Q2_{) are lines through the origin with a slope t ∈ Q the above}

parametization gives an isomorphism of P(Q2) with the rational circle. Therefore, the rational circle is a Brauer-Severi variety.

Another example is C = {(x, y) ∈ Q2 _{| x}2_{+ y}2_{+ 1 = 0}. This equation has no solutions}

in Q, so seems to be a bit boring, but if we extend our scalars to C we do get solutions. We denote C_C= {(x, y) ∈ C2 | x2_{+ y}2_{+ 1 = 0}. Using those solutions we can do exactly}

the same trick as for the circle; look at the intersections of C_C with lines of slope t ∈ C through (i, 0). Assigning to such a line the second intersection point with C_C gives a parametrization of C_C and therefore an isomorphism with P(C2). This is an example of a non-trivial Brauer-Severi variety; we really need to extend our scalars to C to get an isomorphism with projective space.

In this thesis we look at examples of bundles Brauer-Severi varieties. This means that we look at a family of Brauer-Severi varieties parametized by a given base B. So, to each b ∈ B we assign a Brauer-Severi variety BS(b) in a compatible way. In particular, we are interested in non-trivial examples such as the second one we gave, since not many explicit examples are known.

(30)

Bibliography

[Aue+17] Asher Auel et al., eds. Brauer groups and obstruction problems. Vol. 320. Progress in Mathematics. Moduli spaces and arithmetic. Birkh¨auser/Springer, Cham, 2017, pp. ix+247.

[AW14a] Benjamin Antieau and Ben Williams. “The topological period-index problem over 6-complexes”. In: J. Topol. 7.3 (2014), pp. 617–640.

[AW14b] Benjamin Antieau and Ben Williams. “Unramified division algebras do not always contain Azumaya maximal orders”. In: Invent. Math. 197.1 (2014), pp. 47–56.

[DF84] Frank R. DeMeyer and Timothy J. Ford. “On the Brauer group of surfaces”. In: J. Algebra 86.1 (1984), pp. 259–271.

[Gir71] Jean Giraud. Cohomologie non ab´elienne. Tech. rep. Die Grundlehren der mathematischen Wissenschaften, Band 179. 1971, pp. ix+467.

[Gro95] Alexander Grothendieck. “Le groupe de Brauer. I. Alg`ebres d’Azumaya et interpr´etations diverses”. In: (1995), Exp. No. 290, 199–219.

[GS17] Philippe Gille and Tam´as Szamuely. Central simple algebras and Galois coho-mology. Vol. 165. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2017, pp. xi+417.

[Har77] Robin Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496.

[Huy16] Daniel Huybrechts. Lectures on K3 surfaces. Vol. 158. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2016, pp. xi+485.

[Jon03] Johan de Jong. “A result of Gabber”. In: preprint 25 (2003), pp. 36–57. [Jon04] Johan de Jong. “The period-index problem for the Brauer group of an

alge-braic surface”. In: Duke Math. J. 123.1 (2004), pp. 71–94.

[JS10] Johan de Jong and Jason Starr. “Almost proper GIT-stacks and discriminant avoidance”. In: Doc. Math. 15 (2010).

[Mil80] James S. Milne. ´Etale cohomology. Vol. 33. Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980, pp. xiii+323.

[Mil98] James S. Milne. “Lectures on ´etale cohomology”. In: Available on-line at http://www. jmilne. org/math/CourseNotes/LEC. pdf (1998).

(31)

[Sta08] Jason Michael Starr. Brauer groups and Galois cohomology of function fields of varieties. Publica¸cões Matemáticas do IMPA. XX Escola de Álgebra. In-stituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2008, p. 111.

[Stacks] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia. edu. 2018.

Examples of Brauer-Severi schemes

MSc Mathematics

Master Thesis