Splitting families in Galois cohomology

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Dipartimento di Matematica Corso di Laurea in Matematica

Tesi di Laurea Magistrale

Splitting families in Galois

cohomology

23 OTTOBRE 2020

Candidato: Relatore:

Mattia Pirani

Prof. Tamás Szamuely

Controrelatore:

Prof. Mattia Talpo

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Introduction

The main objective of this thesis is the construction of splitting families in Galois cohomology. The reference article is that of Demarche and Florence [DF17].

Let k be a field with absolute Galois group Γ = Γ(k). If A is a finite discrete Γ-module, then we can consider the associated Galois cohomology groups, which we denote by Hn_{(k, A)}_{. Demarche and Florence introduced}

the following definition.

Definition 1(Splitting family of varieties). Let e ∈ Hn(k, A), with n ≥ 2. A countable family of k-varieties {Xi}i∈I is a splitting family for e, if it satisfies

the following properties:

• for every i ∈ I, the k-variety Xi is smooth and geometrically integral;

• for every field extension k ⊂ l, e vanishes on l if and only if there exists an Xi, which has an l-point.

The aim of this thesis is to prove the following theorem.

Theorem 2. Let A be a finite discrete Γ-module and let e ∈ Hn_{(k, A), with}

n ≥ 2. Then, there exists a splitting family of varieties for e. Moreover, if n = 2, one variety is enough.

Historically, the theorem above was known in the setting of central simple algebras. Let k be a field and let m be a natural number prime to the characteristic of the field. It is known that the group H2_{(k, µ}

m) classifies

central simple algebras over k whose period divides n, and also Severi-Brauer varieties with the same property, where µm is the set of m-roots of unity over

k. The latter are smooth projective varieties over k that become isomorphic to projective space after a finite base change (cfr. [GS17]).

Theorem 3. Let x ∈ H2(k, µm). There is a Severi-Brauer k-variety X, such

that the following holds: for every field extension k ⊂ l, X has an l-point if and only if x vanishes in H2(l, µm).

If m = 2, then H2_{(k, µ}

m)classifies quaternion algebras over k. Moreover,

the Severi-Brauer variety of Theorem 3 is the associated conic (cfr. [GS17]). Recently, Rost introduced the concept of norm variety (cfr. [Ros02]), which has been used by Voevodsky to prove the Bloch-Kato conjecture.

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INTRODUCTION iii Theorem 4 (Bloch-Kato conjecture). Let k be field and let m be a prime number, which does not divide the characteristic of k. The natural map:

H1(k, µm) ∪ · · · ∪ H1(k, µm) −→ Hn(k, µ⊗nm )

is surjective for all n. Let

s = x1∪ · · · ∪ xn∈ Hn(k, µ⊗nm )

be a pure symbol. The norm variety X(s), associated to s, verifies the following fundamental property: for every field extensions k ⊂ l, s vanishes over l if and only if the l-variety X(s)l has a 0-cycle of degree prime to m.

We note that norm varieties do not require to have a closed point over l to get that s vanishes over l, but something weaker.

Demarche and Florence studied a closely related problem. Indeed, apply-ing Theorem 2 to A = µm, we obtain a result similar to Theorem 3. If n ≥ 3,

then we have to replace a single variety with a countable family, hence we can associate a geometric object to all cohomology classes, instead of only pure symbols.

The tools used to prove the Theorem 2 are two, one geometric and one algebraic.

The geometric tool are torsors. Let G be a k-linear algebraic group, that is, an affine group scheme of finite type over k.

Definition 5 (Torsor). A G-torsor over k is a k-variety P , equipped with a right action of G, such that the natural map P ×kG → P ×kP given by

(p, g) 7−→ (p, p · g) on the functors of points, is an isomorphism.

We are interested in the case in which G is smooth. Indeed, in this setting, the following proposition holds.

Proposition 6. There exists a bijection {G-torsors over k}

isomorphism

∼

−→ H1(k, G(ks)),

which associates the class of the trivial cocycle to the class of the trivial torsor. The algebraic object is given by the set of Yoneda extensions. Let A be an abelian category and let A, B ∈ A. Let YEn

A(A, B) be the set of exact

sequences of the form:

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INTRODUCTION iv An arrow E → F between two such exact sequences is a commutative diagram:

0 B E1 · · · En A 0

0 B F1 · · · Fn A 0.

id id

We can define an equivalence relation between E, F ∈ YEn

A(A, B). We say

that E is equivalent to F if and only if there exists G ∈ YEn

A(A, B)and a

commutative diagram like the following:

E G F.

Definition 7(Yoneda extensions set). We define the quotient YExtn_A(A, B) := YEn

A(A, B)/ ∼ to be the set of Yoneda extensions.

If the abelian category A has enough injective objects and / or projectives, then we can define the functor Extn_{. The advantage of defining the set of}

Yoneda extensions is that we can extend the functor Extn _{to all abelian}

categories. Indeed, it can be shown that there is a natural isomorphism between Extn_{(A, B)} _{and YExt}n

A(A, B). Moreover, we have the following

theorem.

Theorem 8. Let YExtn_Γ,d_{(Z/dZ, B) be the Yoneda extensions set over the} category of finite discrete Γ-module of d-torsion. There exists a natural isomorphism:

YExtn

Γ,d(Z/dZ, B) → Hn(Γ, B),

where Γ acts on Z/dZ trivially.

Thanks to the latter theorem, we can reformulate Theorem 2 in terms of Yoneda extensions. We set E to be an exact sequence who represents e.

Now, we provide a sketch of proof of the main theorem (in the case n = 2). Let φ : H → G be a morphism of linear algebraic k-groups.

Definition 9 (Lifting triangle). Let X be a k-scheme. A lifting triangle, relative to φ, is a commutative triangle t:

Q P

X

f

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INTRODUCTION v Let P be a G-torsor over k. There is a natural notion of pull-back of lifting triangle, since the involved objects are stable for base change. So, we can give the following definition.

Definition 10 (Lifting variety). A k-variety X is called a lifting variety, for the pair (φ, P ), if there exists a lifting triangle T :

Q P ×kX

X

F

such that the following property holds: for every field extension k ⊂ l and for every lifting triangle t:

Q P ×kSpec l

Spec l

f

there exists an l-point x : Spec l → X, such that the pullback x∗_{(T )} _is

isomorphic to t as a lifting triangle over Spec l.

Theorem 11. Let k be a field, φ : H → G be morphism of linear algebraic k-groups and P be a G-torsor over k. There exists a lifting variety X, for the pair (φ, P ). Moreover if G is smooth and φ is surjective, then X is geometrically integral and smooth.

The most important property of lifting varieties is given by the following proposition.

Proposition 12. Let X be a lifting variety, for the pair (φ, P ), then the following property holds: for every field extension k ⊂ l, the k-variety X has an l-point if and only if P lifts, through φ, to an H-torsor over l.

Now, we introduce the concept of E-diagram, where E is an element of YEn

A(A, B). Our aim is to use E-diagrams to reduce the proof of Theorem 2

to the construction of a lifting variety as in Proposition 12.

Definition 13 (E-diagram). An E-diagram is a couple (Y , φ) such that: • Y ∈ YEn−1

A (En, B).

• φi : Ei → Yi is a family of arrows making the following diagram

commute:

0 B E1 · · · En A 0

0 B Y1 · · · En 0

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INTRODUCTION vi The previous definition is motivated by the following proposition. Proposition 14. Let A be an abelian category and let A, B ∈ A. Let E ∈YEnA(A, B). Then, there exists an E-diagram if and only if the class of

the sequence E is trivial in YExtn_A(A, B).

In particular, we focus on those cases where A is either the category of finite discrete Γ-modules of d-torsion or the category of finite Z/dZ-modules. We denote those categories by MΓ,d (or Mk,d) and Md, respectively.

Proposition 15. An extension E ∈YEn_k,d(A, B) (an E-diagram Y ) is equiv-alent to E0 ∈YEn_d(A, B) (an E0-diagram Y0) and a continuous group mor-phism c : Γ →Aut E0 (Γ →Aut Y0), whereAut E0 (Aut Y0) has the discrete topology.

Let E ∈ YEn

k,d(A, B) and let (Y0, φ0) be an E-diagram in Md. There

exists a canonical group morphism:

Aut Y0_→_{Aut E}0_,

where Aut Y0 _{and Aut E}0 _{mean, respectively, the group of automorphism of}

(Y0_{, φ}0_{) and E}0_.

Proposition 16. In our setting, the E-diagram (Y0, φ0) always exists in Md.

Moreover, if n = 2, then there is exactly one E-diagram, up to isomorphism. Proposition 17. Let k ⊂ l be a field extension. Then, there exists a contin-uous lift

Aut Y0

Γ(l) Γ Aut E0 if and only if Y0 has structure of E-diagram in M_l,d.

We equip Aut Y0 _{and Aut E}0 _{with a trivial action of Γ. Then, we can}

reformulate Theorem 2: if [c] is the class of a cocycle in H1_(k,_{Aut E}0₎_{, then}

we want to study on which fields k ⊂ l, [c] is in the image of H1(l,Aut Y0) −→ H1(l,Aut E0).

If we consider the finite constant group schemes associated, then we can study this problem in terms of torsors, by Proposition 6. We can conclude applying Proposition 12.

This thesis is divided in four chapters. In Chapter 1, we introduce the functorial point of view in algebraic geometry and some useful facts about Abelian categories.

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INTRODUCTION vii In Chapter 2, we develop the theory of group schemes, concentrating on the parts of interest to our aim: linear algebraic groups and torsors.

In Chapter 3, we develop the theory of Yoneda extensions. We will see, in particular, Yoneda extensions in the category of finite discrete Γ-modules of d-torsion.

In Chapter 4, we see the proof of the main theorem of the thesis. In the first part we introduce the lifting varieties and in the second one the E-diagrams.

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Chapter 1

Preliminary notions

1.1 The functorial point of view in algebraic

geom-etry

In this section we are going to discuss a way of embedding the category of schemes into the larger category of contravariant functors F from the category of schemes to the category of sets. This embedding is useful in at least two ways:

• The effect of some basic constructions, such as products, is much easier to describe on functors of points than on schemes.

• In trying to construct a certain scheme, it is often easy to construct the functor that would be the functor of points of that scheme, if the scheme existed; the construction problem is then reduced to the problem of proving that the functor is representable and the use of Yoneda’s Lemma.

Recall the statement of Yoneda’s Lemma (cfr. [Mac98, Section III.2]): Theorem 1.1.1 (Yoneda lemma). Let C be a category. The functor

C −→Fun(Cop_, _Set)

X 7−→HomC(−, X)

is fully faithful.

Remark 1.1.2. We will indicate the functor HomC(−, X)as hX or, when it is

clear from the context, as X, too.

Fix a scheme S. We work, most of the time, in the setting C = Sch/S, which is the category of schemes over S.

When S = Spec R is affine, we can say something more. For a proof of the next theorem see [EH00, Proposition VI.2]:

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CHAPTER 1. PRELIMINARY NOTIONS 2 Theorem 1.1.3. Let C be the category of R-algebras. The functor:

C −→Fun(C, Set) X 7−→HomC(X, −)

is fully faithful. Representable functors are affine schemes over R. 1.1.1 Grothendieck topologies

Grothendieck took one step further by observing that sometimes one does not even need to know the open subsets: for a lot of purposes (for instance, the concept of a sheaf), it suffices to have a notion of open covering. This led to the notion of a Grothendieck topology, which is usually not a topology in the standard sense.

Good references for the following arguments are [Mil80] and [Vis04]. Definition 1.1.4 (Grothendieck topology). Let C be a category. We con-sider all families of morphisms {Ui → U }i∈I having a common target. A

Grothendieck topology on C is a set T whose elements are some of these families, called open coverings, satisfying the following axioms:

• (Isomorphisms are open coverings) If U0 _{→ U} _{is an isomorphism, then}

the one-element family {U0 _{→ U }} _{belongs to T .}

• (An open covering of an open covering is an open covering) If {Ui → U }

belongs to T and {Vij → Ui}belongs to T for each i, then {Vij → U }

belongs to T .

• (A base extension of an open covering is an open covering) If {Ui → U }

belongs to T and V → U is a morphism, then the fiber products V ×UUi exist and {V ×UUi → V } belongs to T .

Definition 1.1.5 (Site). A pair (C, T ), consisting of a category C and a Grothendieck topology T on C is called a site.

Example 1.1.6. A topological space (X, τ) induces a site in a natural way. Let’s see some useful sites. Fix a scheme S:

• (The Zariski site) It is the site associated to the underlying topological space of S. It is indicated as SZar.

• (The étale site) Let C be the category whose objects are schemes U equipped with an étale morphism U → S and whose morphisms are S-morphisms. Call a family {φi : Ui → U }of morphisms in C an open

covering if S φi(Ui) = U as topological spaces. This defines the étale

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CHAPTER 1. PRELIMINARY NOTIONS 3 • (The flat site) Let C be the category Sch/S. An open covering is a family {φi : Ui → U }of S-morphisms such that ` Ui → U is faithfully

flat and of finite type. This defines the flat site Sflat.

Remark 1.1.7. Interest in the flat topology resides in the fact that faithfully flat morphisms of finite type have good descend property (cfr. [Gro67, Section 17.7]).

1.1.2 Presheaves and sheaves

Definition 1.1.8 (Presheaf). A presheaf F (of sets) on a category C is a functor

Cop →Set

An element of F (U) is called a section of F over U. Similarly define presheaves of groups, abelian one, ecc... A morphism of presheaves is a morphism of functors.

Definition 1.1.9 (Sheaf). Let F be a presheaf on a site (C, T ). We say that F is a sheaf if the following diagram:

F (U ) →Y i F (Ui) ⇒ Y i,j F (Ui×UUj)

is an equalizer for all open coverings {Ui → U }in T . A morphism of sheaves

is a morphism of presheaves.

Example 1.1.10. Let F and G be sheaves of sets on the site (C, T ). Consider the presheaf:

H : U −→Hom(F|U, G|U).

The presheaf H actually is a sheaf.

The following fundamental proposition holds, see [Poo17, Proposition 6.3.19] for a proof.

Proposition 1.1.11. Let S be a scheme, and let X be an S-scheme. The functor of points h_X, viewed as a presheaf on the flat site S_flat, is a sheaf. Moreover, the same is true for the Zariski and the étale sites.

Hence, by Yoneda’s Lemma, there is an equivalence of categories between schemes and representable sheaves.

Definition 1.1.12(Sheafification of a presheaf). A sheafification of a presheaf F is a sheaf F+ equipped with a morphism i : F → F+, such that every presheaf morphism F → G factors uniquely through i.

Fortunately the sheafification always exists, see [Vis04, Theorem 2.64] for a proof.

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CHAPTER 1. PRELIMINARY NOTIONS 4 Definition 1.1.13 (Sheaf quotient). Let G be a group sheaf which acts on a sheaf F . We can consider the presheaf:

F/G : U −→ F (U )/G(U ).

We define the sheaf quotient of F by G to be the sheafification of F/G. Remark 1.1.14. Sheaf quotients can be characterized by an appropriate universal property. Indeed, they are a categorical quotient (cfr. [MFK94, Definition 0.5]).

1.2 Abelian categories

In this section we are going to study some general facts about abelian categories. We will also focus on some specifics abelian categories.

All the categories considered are supposed to be abelian.

Definition 1.2.1 (Complete sub-category). Let A be a full sub-category of B. We say that A is a complete sub-category of B if the following condition holds: for all exact sequences in B:

B1 → B2 → B3 → B4 → B5,

if B1, B2, B4, B5 ∈ A, then B3 ∈ A, too.

Proposition 1.2.2. Let A be a complete sub-category of B, the inclusion functor is an additive exact one.

Sketch of the proof. We have to apply the hypothesis to appropriate exact sequences. To prove that the inclusion is an additive functor, we take the following exact sequences in B:

0 −→ A −→ A ⊕ B −→ B −→ 0.

By hypothesis, A ⊕ B is the direct sum of A and B in A. To prove that the inclusion preserve kernels, we take the following exact sequences in B:

0 −→ K −→ A −→ B. By hypothesis, K is the kernel of A → B in A.

Example 1.2.3. Let B be a category of modules and A be the full sub-category of finite ones. Hence A is a complete sub-category of B.

In abelian categories pullback and pushout always exist, indeed they can be expressed as appropriate kernels and cokernels in the following way:

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CHAPTER 1. PRELIMINARY NOTIONS 5 Proposition 1.2.4. Let f : B → A and g : C → A be morphisms in A. There exists the exact sequence:

0 B ×AC B ⊕ C h A,

where h = (f, −g) and B ×_AC is the pullback of B and C over A.

Dually, let f0: A → B and g0 : A → C be morphisms in A. There exists the exact sequence:

A B ⊕ C B`

AC 0,

h0

where h0 = (f0, −g0) and B`

AC is the pushout of B and C over A.

Proof. Let’s see the first exact sequence, the proof of the second one is dual. Let K be the kernel of h : B ⊕ C → A and let i : K → B ⊕ A the inclusion map. Let p1 : B ⊕ C → B be the projection map and let j1: B → B ⊕ C be

the inclusion one. Analogously, we define p2 and j2. We have to show that

K verifies the universal property of pullback. First of all, we have to show that the following diagram commutes:

K C

B A.

p2◦i

p1◦i g

f

Since i : K → B ⊕ C is the kernel of h, the following equalities hold: 0 = h ◦ i = h ◦idB⊕C ◦ i = h ◦ (j1◦ p1+ j2◦ p2) ◦ i = f ◦ p1◦ i − g ◦ p2◦ i,

as we want to prove. Suppose that the following diagram exists: Z

K C

B A,

g f

we have to show that there is a unique map Z → K making commute the diagram. This follows by the universal property of kernels. Indeed, as we have seen before, the existence of the big commutative square implies the existence of a morphism Z → B ⊕ C, such that the composition with h is the zero map.

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CHAPTER 1. PRELIMINARY NOTIONS 6 Proposition 1.2.5. Let

A1 A2 A3 A4 A5

be an exact sequence. Let φ₄ : B4 → A4 be a morphism. There exists the

following exact commutative diagram:

A1 A2 A3×A4 B4 B4 A5

A1 A2 A3 A4 A5.

id id φ4 id

Dually, let ψ2 : A2 → B2 be a morphism. There exists the following exact

commutative diagram:

A1 B2 B2`_A₂A3 A4 A5

A1 A2 A3 A4 A5.

id ψ2 id id

Sketch of the proof. Let’s see the first sentence, since the second one is dual. Without loosing of generality we can suppose that A1 = 0. We have to

construct the morphism A2 → A3×A4 B4. By Proposition 1.2.4, the fiber

product A3×A4B4 is the kernel of h : A3⊕ B4 → A4. There is the canonical

map A2 → A3⊕ B4, which composes with h is the zero morphism. Then,

by universal property of kernels, the map factorizes through A3 ×A4 B4.

Moreover A2 → A3×A4B4 is a monomorphism, since A2→ A3⊕ B4 is. Let’s

see the exactness in A3 ×A4 B4. Let K be the kernel of A3×A4 B4 → B4.

By hypothesis there is K → A2, which makes the diagram commute. By

construction, the composition A2 → A3×A4 B4 → B4 is zero, then there is a

morphism A2 → K, too. By universal properties of objects involved, these

maps are mutual inverse, hence we have the thesis. 1.2.1 _{Finite Z/dZ-modules}

In this section we are going to study the category of finite Z/dZ-modules, which we denote by Md. Some properties that we prove can be generalized

to quasi-Frobenius rings (cfr. [NY03]).

The category Mdis abelian, with enough projectives, since A is noetherian

ring. By the structure theorem for finitely generated abelian groups, every A ∈ Mdis isomorphic to a finite sum:

A 'M

i

Z/diZ , where di divides d, for all i.

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CHAPTER 1. PRELIMINARY NOTIONS 7 Proposition 1.2.6. Every free module is an injective object in Mdand every

object is included in a free module. In particular M_d has enough injectives. The proof of the Proposition 1.2.6 is a direct consequence of the following one.

Proposition 1.2.7. The functor F :=Hom(−, Z/dZ) : Mop_d → Md is exact

and F ◦ F is naturally equivalent to the identity functor.

The module ˆA := F (A) is said to be the dual of A. The following proposition holds, the proof is obvious.

Proposition 1.2.8. The dual of a free module is still a free module and the cardinalities of A and ˆA are equal.

Proof. (Proposition 1.2.6) Let A ∈ Mdand let P → ˆAbe a free module that

maps surjectively on ˆA. We can apply the dual functor, obtaining A → ˆˆˆ P. By Proposition 1.2.7, the map is injective and _{A ' A. By Proposition 1.2.8,}ˆˆ

ˆ

P is free, hence the proof is complete.

Proof. (Proposition 1.2.7) Let Φ : A →Aˆˆ be the natural morphism: a −→ (γ 7→ γ(a)).

By structure theorem, for every a, we can construct a morphism γasuch that

γa(a) 6= 0, hence Φ is injective. Moreover, by Proposition 1.2.8, A and Aˆˆ

have the same cardinality, then Φ is an isomorphism, too. Let 0 −→ A−→ Bf −→ C −→ 0g

be an exact sequence. If we apply F , then we get ˆ

C−→ ˆgˆ B −→ ˆfˆ A −→ 0.

The map ˆg is injective if and only if the cardinalities of Im ˆg and ˆC coincide if and only if | ˆB| = | ˆA| · | ˆC|. By exactness of the first exact sequence there is:

| ˆB| | ˆA| =

|B|

|A| = |C| = | ˆC| . Then the proof is concluded.

The following proposition is a result of rigidity, which is useful in section 4.2:

Proposition 1.2.9. Let φ, ψ : E → M be injective maps of Z/dZ-modules, such that M is free and finite. There is an automorphism : M → M such that ψ = ◦ φ.

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CHAPTER 1. PRELIMINARY NOTIONS 8 The proof is a consequence of Bass’ Theorem (cfr. [Lam91, Theorem 20.9]), which we present in a weaker version:

Lemma 1.2.10 (Bass’ Theorem). Let R be a finite (not necessarily commu-tative) ring. Let J be a left ideal of R and let h ∈ R. If Rh + J = R, then

h + J contains an unit of R.

Proof. (Proposition 1.2.9) Define A := ψ(E), the goal is to extend φ ◦ ψ−1 : A → M to an automorphism of M. By Proposition 1.2.6, M is injective, then we can extend φ ◦ ψ−1 _{to a morphism h : M → M. Let R be the finite}

ring HomZ/dZ(M, M ). Let J be the left ideal {f ∈ R| f|A= 0}. It’s enough

to show the identity

R = Rh + J,

indeed, by Lemma 1.2.10, there exists a unit in h + J. In particular extends φ ◦ ψ−1_{, as we wanted to show. If we show that id ∈ Rh + J, then}

the identity will be satisfied because Rh + J is a left ideal. By construction, the intersection ker h ∩ A is zero. We can consider the following map:

g : ker h ⊕ A → M (b, a) −→ b .

Since M is an injective object, we can extend g to all M and, by construction, g ∈ J. Define f := id − g, if we show that f factorizes through h, then we can conclude. By construction, f| ker h ≡ 0, so we can construct the following

commutative diagram: M M M/ ker h M , h f ˆ h ˆ f

where ˆh is injective. Since M is an injective object, we can extend ˆf to a map f0 _{: M → M}_{. Then, id − g = f = f}0_{◦ h}_{, which concludes the proof.}

We present some useful results about exact sequences and free modules in Md.

Proposition 1.2.11. Let’s consider a short exact sequence in M_d:

0 F1 F2 F3 0 .

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CHAPTER 1. PRELIMINARY NOTIONS 9 Proof. If F3 is free, we get an isomorphism F2 ' F1⊕ F3. In particular, the

isomorphism is of abelian groups, then we can apply the structure theorem. By the uniqueness of the decomposition, we get that F1is free if and only if F2

is free. If F1is free, then, by Proposition 1.2.6, F1is injective and consequently

there is an isomorphism F2 ' F1⊕ F3. We conclude as before.

The previous Proposition can be generalized to longer exact sequence in the following way:

Proposition 1.2.12. Let’s consider an exact sequence in Md:

0 F1 F2 F3 · · · Fn 0 .

If all modules are free except one, then all modules are free.

Proof. We split the sequence in short exact ones. Applying many times Proposition 1.2.11, we get the claim.

1.2.2 Finite discrete Γ-modules of d-torsion

In this section we are going to study the abelian category of finite discrete Γ-modules of d-torsion, where Γ is a profinite group. We indicate this category as MΓ,d.

Recall from [Ser02, Section 2.1], that a Γ-module A is discrete if the stabilizer of each elements of A is an open subgroup of Γ, or equivalently, if the identity A = S AU _{holds, where U runs over all open (normal) subgroups}

of Γ.

Proposition 1.2.13. If A is a finite discrete Γ-module, then there exists an open (normal) subgroup U of Γ, such that A = AU.

Proof. One inclusion is obvious, for the other one it is enough to take the intersection of all stabilizers.

Proposition 1.2.14. Let A be a finite Γ-module. If we give toAut_Z(A) the discrete topology, then the following holds: A is a discrete Γ-module if and only if the natural map Γ →Aut_Z(A) is continuous.

Proof. ⇐) If U is the kernel of Γ → Aut_Z(A), then A = AU and, by the continuity hypothesis, U is open.

⇒) It is enough to show that the kernel U is an open set. Since we can write U =T

a∈AStab (a), we get the thesis.

Proposition 1.2.15. Let A be a discrete Γ-module of d-torsion. A is finite if and only if is finitely generated.

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CHAPTER 1. PRELIMINARY NOTIONS 10 Proof. ⇒) It’s obvious.

⇐) Let {a1, . . . , ak} be a set of generators for A. Since A is discrete, the

orbit of ai is finite, for all i = 1, . . . , k. Since A is finitely generated, as

Z/dZ-module, by {orb(a1), . . . ,orb(ak)}, then it is finite.

Proposition 1.2.16. Let A ∈ M_Γ,d. There exists F ∈ M_Γ,d_{, free as} Z/dZ-module, and a surjective Γ-equivariant group morphism: φ : F → A.

Proof. Let F be the finite free Z/dZ-module L_a∈A_{aZ/dZ. We give to F} structure of Γ-module in the following way:

γ · X a∈A a · ea ! 7−→ X a∈A γ(a) · ea .

The action is discrete, indeed if U is an open (normal) subgroup such that A = AU, then F = FU. By construction there is a surjective Γ-equivariant group morphism F → A: X a∈A a · ea7−→ X a∈A ea a.

We want to construct a dual functor, like as in the category Md. We

have to give a discrete Γ-module structure to ˆA :=Hom(A, Z/dZ): (γ, φ) −→ (γ · φ : a → φ(γ · a)).

Since A is discrete, ˆAis discrete, too. Moreover, if f : A → B is a morphism in MΓ,d, then the dual ˆf is Γ-equivariant, too. Then it makes sense to state:

Proposition 1.2.17. The functor F := Hom(−, Z/dZ) : Mop_Γ,d → M_Γ,d is exact and F ◦ F is naturally equivalent to the identity.

Proof. The natural morphism Φ : A →Aˆˆ is Γ-equivariant. Indeed, for all ψ ∈ ˆA, we have:

Φ(γ · a)(ψ) = ψ(γ · a) = (γ · ψ)(a) = Φ(a)(γ · ψ) = (γ · Φ(a))(ψ). Then we conclude as in Proposition 1.2.7.

We give the following propositions, which proofs follow those already seen in the previous subsection.

Proposition 1.2.18. Every object in MΓ,d can be embedded in a finite

Γ-module of d-torsion, which is free as Z/dZ-module.

Proposition 1.2.19. Let’s consider a long exact sequence in M_Γ,d:

0 F1 F2 F3 · · · Fn 0 .

Suppose that all modules are free as Z/dZ-modules except one, then all modules are free.

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CHAPTER 1. PRELIMINARY NOTIONS 11 1.2.3 Discrete Γ-modules of d-torsion

In this subsection we are going to study the abelian category of discrete Γ-modules of d-torsion, where Γ is a profinite group. We indicate this category as B.

Lemma 1.2.20. Let I be an injective object in the category of discrete Γ-modules. The d-torsion part of I, say Id, is an injective object in B.

Proof. We have to verify the universal property. Let A → B be a monomor-phism in B. Let A → Id be a morphism, we have to lift the map in the

following way:

A B

Id

I

By injectivity of I, there is a lift B → I. Since B is a d-torsion module, the morphism factorizes through Id, then we get the thesis.

Then we can prove the following proposition:

Proposition 1.2.21. The category B has enough injectives.

Proof. We construct injective objects in B starting from those of the category of Γ-modules. Let M be a discrete Γ-module of d-torsion. Recall by [Mac67, Theorem 7.4] that M can be embedded in an injective Γ-module I. Let I0 _be

[

U

IU,

where U ranges over all open (normal) subgroups of Γ. By the same argument as in Lemma 1.2.20, I0 _{is an injective object in the category of discrete}

Γ-modules. Moreover, since M is discrete, the morphism M → I factorizes through I0_{. By Lemma 1.2.20, I}0

dis an injective object in B. Moreover, the

map M → I0 _{factorizes through I}0

d, then we get the claim.

Then it makes sense to define the Ext functor over B.

Proposition 1.2.22. MΓ,d is a complete sub-category of B. Every object in

B can be written as an inductive limit of elements of M_Γ,d.

Proof. The first claim is obvious. The second one is not difficult. Indeed every discrete Γ-module of d-torsion can be written as a union of finitely generated Γ-module of d-torsion. We conclude applying Proposition 1.2.15.

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CHAPTER 1. PRELIMINARY NOTIONS 12 Remark 1.2.23. The category B has the AB5 property, i.e. B admits direct sums, inductive limits exists and the inductive limit functor is exact. Indeed the inductive limits can be expressed as a quotient of the direct sum by an appropriate equivalence relation.

Since the AB5 property holds, there is the following proposition, which is a particular case of [Ser60, Proposition 6]:

Proposition 1.2.24. Let C and D be in B. If C can be written as an inductive limit

lim

→ Ci= C,

then the natural morphism lim

→ Ext

n

B(D, Ci) →ExtnB(D, C)

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Chapter 2

Group schemes

In this chapter we are going to develop a little bit of the theory of group schemes. In particular, we are going to focus on linear algebraic groups and torsors. Most of the material presented comes from the standard references [Mil80] and [Wat79].

Definition 2.0.1. (Group scheme) Let S be a scheme and let G be a scheme over S. We say that G is a group scheme over S if there are the following morphisms of S-schemes: • Multiplication map: m : G ×SG → G. • Inverse map: i : G → G. • Identity map: e : S → G. Such that the following natural diagrams commute:

• Associativity: G ×SG ×SG G ×SG G ×SG G. m×id id×m m m • Inverses: G G ×SG G G i×id e m id×i e 13

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CHAPTER 2. GROUP SCHEMES 14 • Identity: G G ×SG G G . e×id id m id×e id

Remark 2.0.2. Let S0 → S be a morphism of schemes. The base change G ×SS0 inherit a natural structure of group scheme over S0.

Definition 2.0.3 (Morphism of group schemes). Let G and G0 be group schemes over S. A morphism φ : G → G0 _{of group schemes is a morphism of}

schemes over S, which commutes with the multiplication map.

Recall from section 1.1 the functorial point of view in algebraic geometry. Hence a group scheme over S is the same as a representable functor, which factorises:

(Sch/S)op Set

Grp,

U

where U is the forgetful functor.

The following definition arises spontaneously:

Definition 2.0.4. (Group action) Let S be a scheme, let X be a scheme over S and let G be a group scheme over S. We say that G acts (on the right) on X over S if there exists a morphism of S-schemes:

σ : X ×SG −→ X,

such that the natural diagram commutes:

X ×SG ×SG X ×SG

X ×SG X.

σ×id

id×m σ

σ

We say that X is a G-object.

Remark 2.0.5. Let G be a group scheme over S, acting on the left on X. We can define a right action of G on X in the following way: for all S-schemes U, there is G(U) × X(U) → X(U) such that

(g, x) 7−→ x · g−1.

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CHAPTER 2. GROUP SCHEMES 15

2.1 Linear algebraic k-groups

In this section we will investigate some geometric properties of groups. All groups considered are linear algebraic over k, that is an affine k-group scheme of finite type.

Definition 2.1.1 (Hopf algebra). Let A be a k-algebra. We say that A is an Hopf algebra, if there are the following k-morphisms:

• Comultiplication map: ∆ : A → A ⊗ A. • Coinverse map: j : A → A. • Coidentity map: : k → A. Such that the following natural diagrams commute:

• Coassociativity: A ⊗ A ⊗ A A ⊗ A A ⊗ A A ∆⊗id id⊗∆ ∆ ∆ • Coinverse: A A ⊗ A A A id⊗j j⊗id ∆ • Coidentity: A A ⊗ A A A . id⊗ ⊗id ∆ id id

Remark 2.1.2. Applying Theorem 1.1.3, we get that linear algebraic k-groups coincide with functors

k-Alg −→ Grp,

which are represented by a finitely generated Hopf algebra over k.

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CHAPTER 2. GROUP SCHEMES 16 • (Roots of unity µn) Let µn be the functor represented by k[t]/(tn− 1).

For all k-algebras R, the set

Homk(k[t]/(tn− 1), R) = {x ∈ R| xn= 1}

has a natural commutative group structure by the multiplication of R. Hence µn defines a linear algebraic k-group.

• (Finite constant groups) Let G be a finite group. The finite constant group associated to G is given by the following Hopf algebra’s structure on kG_: ∆ : 1g 7−→ X h·k=g 1h⊗ 1k, j : 1g 7−→ 1g−1, : 1e7−→ 1,

where 1gindicates the element of kG, which has an 1 in the g-component

and 0 in the other ones. 2.1.1 Linear representations

Let V be a k-vector space. We can define the set functor k-Alg → Set: V : R 7−→ V ⊗kR.

Analogously, we can define the group functor: GLV : R 7−→AutR(V ⊗kR).

If V = kn_{, then both functors are linear algebraic k-groups. Indeed, V is}

represented by k[x1, · · · , xn]:

kn⊗_kR = Rn=Homk(k[x1, · · · , xn], R).

The functor GLV is represented by k[x11, x12, · · · , xnn, y]/(y · det −1), where

detis the determinant, indeed

AutR(kn⊗kR) =AutR(Rn)

is given by n × n matrices with coefficient in R and invertible determinant, and then

AutR(Rn) =Homk(k[x11, x12, · · · , xnn, y]/(y · det −1), R).

If V is finite-dimensional, then we fix a basis and we construct an isomorphism with kn_{. This isomorphism induces ones between V and k}n_{, GL}

V and

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CHAPTER 2. GROUP SCHEMES 17 Definition 2.1.4 (Linear representation). Let G be a linear algebraic k-group and let V be a k-vector space. A linear representation of G on V is a group morphism:

G −→GLV.

If V is finite dimensional, then we say that the representation is (dim V )-dimensional.

Remark 2.1.5. A linear representation of G on V is equivalent to an action of G on V .

Definition 2.1.6 (Comodule). Let V be a k-vector space and let A be an Hopf algebra over k. We say that V is an A-comodule, if there is a k-linear morphism:

ρ : V −→ V ⊗kA,

such that the following natural diagrams commute: • V V ⊗kA V ⊗kA V ⊗kA ⊗kA ρ ρ _id⊗∆ ρ⊗id • V ⊗kA V V . id⊗ ρ id

Let W be a k-vector subspace of V . We say that W is an A-subcomodule of V if ρ(W ) ⊂ W ⊗ A.

Remark 2.1.7. If W is an subcomodule of V , then ρ restricts to an A-comodule structure on W .

Remark 2.1.8. Let A be a Hopf k-algebra. The comultiplication ∆ induces an A-comodule structure on A.

The following propositions hold, for a proof see [Wat79, Section 3.2] and [Wat79, Section 3.3], respectively.

Proposition 2.1.9. Let G a linear algebraic k-group represented by the Hopf algebra A. A linear representations of G on V is equivalent to an A-comodule structure on V .

Proposition 2.1.10. Let A be a Hopf k-algebra. Every A-comodule V is a directed union of finite-dimensional subcomodules.

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CHAPTER 2. GROUP SCHEMES 18 We can prove that every linear algebraic k-group is a matrix one. Proposition 2.1.11. Let G be a linear algebraic k-group. There is a closed embedding in some GLn.

Sketch of the proof. Let A be a the Hopf finitely generated k-algebra, which represents G. By Proposition 2.1.9, the comultiplication ∆ of A gives a linear representation of G on A. By Proposition 2.1.10, there is a finite dimensional subcomodule V of A, such that V contains the generators of A, since they are in a finite number. Let {vj}be a basis of V , and write ∆(vj) =P vi⊗ aij.

The aij are the matrix entries (image of xij) in the corresponding map of G

to GLn. Hence the morphism

k[x11, x12, · · · , xnn, y]/(y · det −1) −→ A,

associates aij to xij. Since there is

vj = ( ⊗id)∆(vj) =

X

(vi)aij,

the image of the above morphism contains all the vj, hence the morphism

is surjective. Therefore the induced morphism between schemes is a closed embedding.

Definition 2.1.12 (Faithful representation). Let G be a linear algebraic k-group and let φ : G → GLV be a linear representation of G on V . If

φis a closed embedding we say that the linear representation is a faithful representation.

Then, Proposition 2.1.11 shows that:

Proposition 2.1.13. Let G be a linear algebraic k-group. There is a vector space V and a faithful linear representation of G on V .

2.1.2 Quotient schemes

Recall by Subsection 1.1.2, the existence of quotient as flat sheaf, which is a categorical quotient, in the sense that make the appropriate diagram commute and satisfies an universal property (cfr. [MFK94, Definition 0.5]). There is a second type of quotient that we can define, which is the geometric one (cfr. [MFK94]).

Definition 2.1.14 (Geometric quotient). Given an action σ of a group G on a k-variety X, a pair (Y, φ), such that Y is a k-variety and φ : X → Y a morphism of k-schemes, will be called geometric quotient (of X by G) if

• The geometric fibers of φ are precisely the orbits of the geometric points of X.

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CHAPTER 2. GROUP SCHEMES 19 • A subset U ⊂ Y is open if and only if φ−1_{(U )}_{is open in X.}

• The structure sheaf OY is the subsheaf of φ∗(OX)consisting of invariant

functions for the action.

This type of quotient is stronger then the categorical one. Indeed, the following proposition holds, for a proof see [MFK94, Proposition 0.1]. Proposition 2.1.15. Let σ be an action of a group G on a k-variety X and suppose (Y, φ) is a geometric quotient of X by G. Then (Y, φ) is a categorical quotient of X by G, in particular is unique up to isomorphism.

We focus to the following problem. Let H be a closed subgroup of G. We can make H act on G by multiplication. Hence we can ask if G/H exists as scheme over k. There is the following theorem, for a proof see [Bor91, Theorem 6.8].

Theorem 2.1.16. Let G be a linear algebraic k-group and let H be a closed subgroup of G. The geometric quotient G → G/H exists over k. Moreover, G/H is smooth and a quasi-projective k-variety.

Remark 2.1.17. In general G/H is not affine over k. Indeed, there is a one-to-one correspondence between affine quotient and normal subgroup (cfr. [Wat79, Section 16.3]).

In the next section, we will see that some geometric quotients have the good property to be a torsor.

The next proposition is a well known result of Totaro (cfr. [Tot99, Remark 1.4]). This construction will be useful in the last chapter.

Proposition 2.1.18 (Totaro’s construction). Let G be a linear algebraic k-group. There is a finite dimensional vector space V and a closed subset S ⊂ V , such that G acts freely on V −S and the geometric quotient (V −S) → (V − S)/G exists. Moreover (V − S)/G exists as a quasi-projective k-variety. Sketch of the proof. The idea is to reduce to the case of Theorem 2.1.16. By Proposition 2.1.13, there exists a faithful representation W of G, say of dimension n. Let V be the finite dimensional vector space Hom(AN +n_{, W )}_,

where AN +n _{is the (N + n)-dimensional affine space over k and N is a fixed}

non-negative integer. Let S be the closed subset in V of non-surjective linear maps AN +n_{→ W}_{. Fix a decomposition W ⊕ A}N _{of A}N +n_{, hence there is a}

canonical action of GLW on AN +n, which makes G be a closed subgroup of

GLN +n. Let H be the closed subgroup of GLN +n, whose elements make the

following diagram commute:

W ⊕ AN W ⊕ AN

W W,

h

π π

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CHAPTER 2. GROUP SCHEMES 20 where the projection π is the canonical one. First of all, we can construct an isomorphism of k-varieties between V − S and GLN +n/H, indeed it is

enough to show that V − S is the geometric quotient of GLN +n by H. Define

the morphism φ : GLN +n→ V, in the following way:

f 7−→ π ◦ f.

Since π and f are surjective, the morphism maps into the open subset V − S. The action of H on the geometric fibers is free and transitive. Indeed, if f and g are in the same fiber, then f ◦ g−1 _{fixes W . By definition of H, the}

composition f ◦ g−1 _{is into H, moreover it is the unique element of H, which}

sends g into f. To show that the invariant regular functions for GLN +n and

σ, descend to one of V − S, the idea is the following. We can write V − S as union of the affine subset Vm, where m range over all the determinant of

the maximal minors of the matrices Hom(AN +n_{, W )}_{. An invariant regular}

function on GLN +n= (MN +n)det induces a regular function on Vm, indeed,

up to changing coordinates with an element of H, we can suppose that det = mand that f takes values on Hom(AN +n, W ). Moreover these regular functions, built on Vm for all m, merge to an unique regular function on

V − S. Then V −S is the geometric quotient GLn/H. The intersection G∩H

is trivial, moreover H is stable under conjugation for elements of G, hence we can consider the closed subgroup H o G of GLN +n. By construction, the

isomorphism between GLN +n/H and V − S is G-equivariant, then we get

isomorphism

GLN +n/(H o G) → (V − S)/G.

By Theorem 2.1.16, GLN +n/(H oG) exists as quasi-projective k-variety, hence

the same holds for (V − S)/G. Moreover, the map (V − S) → (V − S)/G = GLN +n/H → GLN +n/(H o G) is a geometric quotient. Indeed, it follows

by the fact that GLN +n→GLN +n/H and GLN +n→GLN +n/(H o G) are,

too, by Theorem 2.1.16.

2.2 Torsors

In this section all group schemes will be faithfully flat and of finite-type on the base. Good references for the material proposed are [Mil80], [Poo17] and [Sko01].

Definition 2.2.1 (Torsors). Let G → S be a group scheme. A G-torsor is a scheme P over S, equipped with a right G-action, such that one of the following equivalent conditions holds:

• There exists a covering {Si → S} in the flat topology, such that the

following holds: PSi with its right GSi-action is isomorphic over Si to

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CHAPTER 2. GROUP SCHEMES 21 • The morphism

P ×SG → P ×SP

defined, on functors of points, as (p, g) → (p, p · g), is an isomorphism and P → S is faithfully flat an of finite-type.

A covering, as the previous one, is called a trivializing covering for the torsor P.

See [Mil80, Proposition 4.1] for a proof of the equivalence.

Remark 2.2.2. To be precise the previous definition is about right torsors, there is a dual notion for the left ones. Recall, by Remark 2.0.5, the notation

ˆ

P. It follows by the definition that, if P is a right torsor, then ˆP is a left one, and vice versa.

Remark 2.2.3. Let G be a group scheme, acting on itself by right translation. Take the trivial covering {S → S}, hence, by the first equivalent condition, Gis a G-torsor. It is called the trivial torsor.

Proposition 2.2.4. Let P be a G-torsor. There is a section in P (S) if and only if P is isomorphic, as G-torsor, to the trivial one.

Proof. ⇐) It’s obvious.

⇒) By Definition, there is the isomorphism: P ×SG −→ P ×SP.

Taking the base change by the section in P (S), we get the isomorphism G → P given by:

g 7−→ p · g,

on functors of points. Since it is a G-invariant map, the proof is concluded. Remark 2.2.5. Let S0 → S be a morphism, and let P be a G-torsor over S. The base change PS0 is a G_S0-torsor over S0. Abusing of notation we will say

that PS0 is a G-torsor.

Proposition 2.2.6. Let G be a group scheme over S. Let P1 and P2 be

G-torsors. If P1→ P2 is a G-invariant morphism , then it is an isomorphism.

Proof. We work on functors of points. Taking a trivializing covering for both P1 and P2, we reduce to show that a G-invariant morphism from G to itself,

is an isomorphism. Evaluating G in S we get that a G-invariant morphism from G to itself is given by the left multiplication of a section in G(S). Taking the inverse of this section in G(S), we get the inverse morphism.

Recall, from Proposition 2.1.18, the Totaro’s construction.

Proposition 2.2.7. The scheme morphism (V − S) → (V − S)/G is a G-torsor.

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CHAPTER 2. GROUP SCHEMES 22 Proof. By Proposition 2.1.18, the morphism (V − S) → (V − S)/G is a geometric quotient. By construction, the action of G is free, hence, by [MFK94, Proposition 0.9], the morphism above is a G-torsor.

By descent theory, the following proposition holds, see [Mil80, Proposition 4.2] for a proof.

Proposition 2.2.8. Let G → S be a group scheme and let P be a G-torsor. If G is smooth (étale) over S, then P is smooth (étale) over S.

2.2.1 Torsor sheaves

In this subsection we introduce torsors in the more general setting of sheaves. This point of view has the advantage, for example, of being able to make the quotient. All the sheaves are defined over a site which admits a final object S.

Definition 2.2.9 (Sheaf torsor). Let G be a group sheaf. A G-torsor sheaf P is a sheaf of sets equipped with a right action of G, such that the following holds: there is an open covering {Ui → S}and a family of G|Ui-isomorphisms

P|Ui → G|Ui, where G|Ui acts on itself by right translations.

Remark 2.2.10. Let G be group scheme. There is an equivalence of category between G-torsors and representable G-torsor sheaves defined over the flat site.

Proposition 2.2.11. Let P be G-torsor sheaf. Let P × G → P × P be the following morphism of sheaves: for all U the morphism P (U ) × G(U ) → P (U ) × P (U ) is given by

(p, g) 7−→ (p, p · g). This sheaf morphism is an isomorphism.

Proof. If P = G is trivial. We reduce to that case taking a trivializing covering for P , indeed, being an isomorphism is a local property.

It could happen that not all G-torsor sheaves are representable. Fortu-nately, under appropriate hypothesis, it is true as consequence of flat descend. See [Mil80, Theorem 4.3] for a proof of the next proposition.

Proposition 2.2.12. Let G → S be a group scheme. If the structure mor-phism is affine, then all G-torsor sheaves are representable.

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CHAPTER 2. GROUP SCHEMES 23 Definition 2.2.13 (Contracted product). Let G be a group sheaf. Let F1

and F2be sheaves. If F1 (resp. F2) is equipped with a right (resp. left) action

of G, then there is a natural left action of G on the product F1× F2, given

by g · (f1, f2) = (f1· g−1, g · f2). The sheaf quotient of F1× F2, under this

action of G, is denoted by F1×GF2 and it is called the contracted product

of F1 and F2 with respect to G.

Remark 2.2.14. If F1 has a left action and/or F2 has a right action, we can

change in a right one and/or left one, as in Remark 2.0.5. Then, the way in which G acts on them doesn’t matter.

Also the contracted product exists as a scheme under appropriate hypoth-esis. See [Sko01, Lemma 2.2.3] for a proof of the following proposition. Proposition 2.2.15. Let P be a torsor under a group scheme G over S. Let X be an affine scheme over S, equipped with a left action of G. The contracted product P ×GX exists as an affine S-scheme.

Remark 2.2.16. Let P be a torsor sheaf under a group sheaf G. Let X be a sheaf, equipped with a left action of G. It could happen to call the contracted product as "twist of X by P " and indicate it asP_X_.

There is an useful characterization of twists. Recall from Remark 2.0.5 the notation ˆX and by Example 1.1.10 the Hom-sheaf. The following proposition holds, which is a particular case of [Gir71, Theorem III.1.6.(ii)].

Proposition 2.2.17. Let G be a group sheaf, let P be a G-torsor sheaf and let X be a left G-object. There is a sheaf isomorphism:

α : P ×GX −→HomG(P, ˆX).

Proof. First of all, we construct a sheaf morphism P × X −→HomG(P, ˆX),

hence, for all U, we have to define a functorial map: P (U ) × X(U ) −→HomG|U(P|U, ˆX|U).

Let (p, x) ∈ P (U)×X(U). We have to define a G|U-invariant sheaf morphism

α(p,x) : P|U → ˆX|U. We claim that there is an unique G|U-invariant sheaf

morphism which sends p in x. Since P is a G-torsor, the following isomorphism holds:

P × G−→ P × P.∼ Evaluating in V → U, we get the isomorphism:

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CHAPTER 2. GROUP SCHEMES 24 which sends the couple (p, g) to (p, p · g). Then, for all the couples (p1, p2) ∈

P (V ) × P (V ), there is an unique g ∈ G(V ), such that p2= p1· g. So, we can

define:

α(p,x)(V )(p0) := x|V · g,

where g is the unique element in G(V ), such that p0_{= p}

|V·g. By construction,

α(p,x)is a G|U-invariant sheaf morphism, moreover it is uniquely determined by

the fact that sends p in x. We have to show that the sheaf morphism factorizes to the quotient. Take the element g · (p, x) = (p · g−1_{, g · x) ∈ P (U ) × X(U )}_,

which induces α(p·g−1_{, g·x)}. By definition of the action on ˆX, we get:

α_(p·g−1_{, g·x)}(U )(p) = α_(p·g−1_{, g·x)}(U )((p · g−1) · g) = (g · x) · g = g−1·(g ·x) = x.

By uniqueness, α(p·g−1_{, g·x)} coincides with α_(p,x), then the sheaf morphism

above induces to the quotient:

α : P ×GX −→HomG(P, ˆX).

We have to show that α is a sheaf isomorphism. Since it is a local property, it is enough to show taking an appropriate covering of S. Let {Ui → S}be a

trivializing covering for the torsor sheaf P . Since P|Si is isomorphic to G|Si

as G|Si-torsor, there are the canonical sheaf isomorphisms:

(P ×GX)|Si ' P|Si× G_|Si X|Si ' G|Si× G_|Si X|Si ' X|Si and HomG(P, ˆX)|Si 'HomG(G, ˆX)|Si ' ˆX|Si.

Reading the map α through these isomorphisms, we get the identity one. Then α is an isomorphism.

An application of contracted products is the possibility to define the pushforward of torsors, under a group morphism. Let H → G be a morphism of group shaves. Then G has a natural structure of left H-object.

Proposition 2.2.18. Let P be an H-torsor sheaf. The twist P ×H G has a structure of G-torsor sheaf.

Proof. Take the presheaf

(P × G)/H : U −→ P (U ) × G(U ) H(U ) . We give to P (U )×G(U )

H(U ) a structure of right G(U)-object, in a functorial way:

([p, g], g) −→ [p, gg].

Then (P × G)/H has a structure of G-sheaf, moreover, by universal property of sheafification, the twist P ×H _G_{has it, too. We have to verify that the}

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CHAPTER 2. GROUP SCHEMES 25 twist is locally isomorphic to G, as G-torsors. Let {Si→ S} be a trivializing

covering for P . There are the canonical sheaf isomorphisms: (P ×H G)|Si ' P|Si×

H_|Si _G

|Si ' G|Si.

By construction, these maps are G-equivariant, then the proof is complete. Another tool provided by twisting is the possibility to make torsors trivial under an appropriate group, in the sense that we are going to explain. First of all we have to see a proposition.

Let G be a group sheaf and let P be a G-torsor sheaf. If G acts on itself by inner automorphisms on the left, then it makes sense to consider the twist P ×GG.

Proposition 2.2.19. The twist P ×GG inherits a structure of group sheaf from that of G.

Proof. By definition, the contracted product P ×GGis the sheafification of the following presheaf:

F := (P × G)/G : U −→ (P × G)(U ) G(U ) , where the left action of G(U) on P (U) × G(U) is given by

g · (p, g) := (p · g−1, gg0g−1).

It is enough to define a group structure on the presheaf, indeed, by universal property of sheafification, the twist inherits a group structure, too. Since P is a G-torsor, for every U the action of G(U) on P (U) is free and transitive. We can fix p ∈ P (U) and we can say that every class in (P × G)(U)/G(U) has an unique representative of the form (p, g), for some g ∈ G(U). We define the multiplication map m(U) : F (U) × F (U) → F (U) in the following way:

m(U ) : ([p, g], [p, g0]) 7−→ [p, gg0].

The construction doesn’t depend by the representative chosen. Indeed, if g ∈ G(U ), then we can rewrite [p, g] = [p · g−1, ggg−1] and [p, g0] = [p · g−1, gg0g−1], and so:

[p, gg0] = [p · g−1, ggg0g−1] = m(U )([p · g−1, ggg−1], [p · g−1, gg0g−1]). The construction is functorial in U, then m defines a presheaves morphism. We define the inverse map i(U) : F (U) → F (U) in the following way:

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CHAPTER 2. GROUP SCHEMES 26 The construction doesn’t depend by the representative chosen. Indeed, if g ∈ G(U ), then we can rewrite [p, g] = [p · g−1, ggg−1], and so:

[p, g−1] = [p · g−1, gg−1g−1] = i(U )([p · g−1, ggg−1]).

The construction is functorial in U, then i defines a presheaves morphism. We define the identity element of F (U) to be the class of (p, idG). It is

well defined, since all the elements of this form are in the same class, by transitivity of the G(U)-action on P (U). It remain to show that the group axioms are satisfied. They follows by group axioms of G, let’s see for example that [p, idG]is the identity element for F (U):

m(U )([p, g], [p, idG]) = m(U )([p, idG], [p, g]) = [p, g].

Thanks to the previous proposition, it makes sense to consider torsors under the groupP_{G. The following proposition holds.}

Proposition 2.2.20. Let P and Q be G-torsor sheaves. The twist PQ has structure of PG torsor sheaf, moreover P_{P is isomorphic to the trivial}P

G-torsor sheaf. This map defines a bijection between G-G-torsor sheaves and

P_G-ones.

Sketch of the proof. We proceed as in the proof of Proposition 2.2.19. For all U, we define the following right action:

σ(U ) : (P × Q)(U ) G(U ) × (P × G)(U ) G(U ) −→ (P × Q)(U ) G(U ) , given by ([p, q], [p, g]) 7−→ [p, q · g]. Let g ∈ G(U). The following identities hold:

σ([p, q], [p, g]) = [p, q·g] = [p·g−1, q·g·g−1] = σ([p·g−1, q·g−1], [p·g−1, ggg−1]), then σ is well defined. Taking a trivializing covering for Q, we get a trivializing one for P_Q_{. We define the} P_G_{-isomorphism between} P_P _and P_G _{in the}

following way: f : (P × G)(U ) G(U ) −→ (P × P )(U ) G(U ) , given by [p, g] 7−→ [p, p · g]. Let g ∈ G(U). The following identities hold:

f ([p, g]) = [p, p · g] = [p · g−1, p · g · g−1] = f ([p · g−1, ggg−1]), then f is well defined. Moreover, f is an isomorphism, since the action of G(U )on P (U) is free and transitive.

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CHAPTER 2. GROUP SCHEMES 27 Proposition 2.2.21. Let G be an affine group scheme over S. Let P be a G-torsor over S and let x : S0→ S be a morphism of schemes. The following holds: x∗(P ) is isomorphic to the trivial torsor if and only if x factorizes through P → S.

Proof. By Proposition 2.2.4, a torsor is trivial if and only if it has a section. Consider the following commutative diagram:

S0

x∗(P ) P

S0 S.

id

x

By universal property of fiber product, there is a section S0 _{→ x}∗_{(P )}_{if and}

only if there is a map S0 _{→ P}_{, which make the diagram commute. Hence we}

have proved the claim.

We can say something more:

Proposition 2.2.22. Take the following diagram:

S0 S

S0.

x

id

Let G be a affine group scheme over S0. Let Q be a G-torsor over S0 and let P be a G-torsor over S. The following holds: x∗(P ) is isomorphic to the torsor Q if and only if x factorizes through QP → S.

Proof. Recall that QP is a torsor under QG. By Proposition 2.2.21, x factorizes throughQ_{P → S} _{if and only if x}∗₍Q_{P )} _{is isomorphic to the trivial}

torsorQ_G_{. By Proposition 2.2.20,}Q_Q_{is isomorphic to the trivial torsor} Q_G_.

By functoriality of twist, x∗_{(P )}_{is isomorphic to Q if and only if x}∗₍Q_{P )} _is

isomorphic toQ_Q_{, then the proof is concluded.}

2.2.2 Galois cohomology

We want to study Galois cohomology from a geometrical point of view. Let G be a smooth linear algebraic k-group. We want to prove the following proposition.

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CHAPTER 2. GROUP SCHEMES 28 Proposition 2.2.23. There is a bijection

{G-torsors over k} isomorphism

∼

−→ H1(k, G(ks)),

which associates the class of the trivial cocycle to the class of the trivial torsor. In this setting, torsors have the following useful property.

Proposition 2.2.24. Let P be a G-torsor over k. There is a finite Galois extension k ⊂ l, such that P is the trivial torsor over l.

Proof. By the descent property, see [Gro67, Section 17.7], the torsor P is smooth and of finite type over k. By [Liu02, Proposition 2.20], the set P (ks)

is not empty. In particular, there is a finite Galois extension k ⊂ l, such that P (l) 6= ∅and then P is the trivial torsor over l, by Proposition 2.2.4.

It is enough to prove the following proposition.

Proposition 2.2.25. Let k ⊂ l be a finite Galois extension. There is a bijection

{G-torsors over k, which are trivial over l} isomorphism

∼

−→ H1(Gal(l/k), G(l)), which associates the class of the trivial torsor to the class of the trivial cocycle. Proof of Proposition 2.2.23. It is a direct consequence of Proposition 2.2.24 and Proposition 2.2.25.

Lemma 2.2.26. Let k ⊂ l be a finite Galois extension. There is a natural isomorphism ofGal(l/k)-modules between AutGlGl and G(l).

Sketch of the proof. We prove the Lemma in the setting in which G is the finite constant group scheme associated to a group G. Since G is a finite group scheme, the action on both AutGlGl and G(l) is trivial, hence we can

suppose k = l. Moreover, G(k) = Homk(kG, k) = Gand, if f ∈ AutGG, then

there is the following commutative diagram: kG_⊗

kkG kG⊗kkG

kG _kG_,

f ⊗id

f

where the lateral maps are given by 1g 7→ P_hh0_=g1_h⊗ 1_h0. The map f is

uniquely determined by the image of 1e. Indeed, from one side of the diagram

we get that 1e is sent to Phh0_{=f (e)}1h ⊗ 1h0. From the other side, we get

P

hh0_=e1_{f (h)}⊗ 1_h0. By equating the sums, we get that f sends 1_{f (g)} to 1_{f (e)g}.

Since a map in AutGGis uniquely determined by the image of 1e, we get the

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CHAPTER 2. GROUP SCHEMES 29 Proof of Proposition 2.2.25. We can conclude by an argument of Galois de-scent. Indeed, since G is affine the study of G-torsors which are trivial over l, is equivalent to the study of twisting of the Hopf algebras associated to G. Then, by [GS17, Theorem 2.3.3] there is a base-point preserving bijection between G-torsors over k, up to isomorphism, which are trivial on l and H1(Gal(l/k), AutGlGl). By the previous lemma, we get the claim.

Recall the following Theorem (cfr. [GS17, Example 2.3.4]).

Theorem 2.2.27(Hilbert’s Theorem 90). Let k ⊂ l a finite Galois extension. The Galois cohomology set

H1(Gal(l/k), GLn(l))

is trivial.

An useful application is given by the the following Example.

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Chapter 3

Yoneda extensions

In this chapter, we are going to study Yoneda extensions group, first of all in the general case of an Abelian category, and then working on MΓ,d.

We will mainly refer to articles [Buc59], [Oor64] and [Yon54].

3.1 Introduction to Yoneda extensions

In this section A will indicate an abelian category, n a positive integer and A, B will be object in A.

First of all we have to define the set of extensions:

Definition 3.1.1 (Extensions). An element E ∈ YEn_A(A, B) is an exact sequence:

0 B E1 · · · En A 0.

Example3.1.2. (Trivial extensions) For all n there exists a particular extension, that is called the trivial one:

• If n = 1, then there is:

0 B B ⊕ A A 0.

• If n > 1, then there is:

0 B id B 0 · · · 0 A id A 0.

Given two extensions E, E0_{, a morphism E → E}0 _{is a family of arrows}

{ψ_i}, making the following diagram commute:

0 B E1 · · · En A 0

0 B E₁0 · · · E_n0 A 0.

ψ0 ψ1 ψn ψn+1

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CHAPTER 3. YONEDA EXTENSIONS 31 Remark 3.1.3. Extensions, with the above morphisms, form a category.

Next goal is to define an equivalence relation between extensions and then to consider the quotient of YEn

A(A, B)by this one.

Definition 3.1.4 (Similarity relation). Let E, F ∈ YEn_A(A, B). We say that E and F are similar if there exists one of the following commutative diagram:

0 B E1 · · · En A 0 0 B F1 · · · Fn A 0, id id or 0 B F1 · · · Fn A 0 0 B E1 · · · En A 0. id id

Remark 3.1.5. Two extensions are similar if there exists a morphism between them, such that ψ0 and ψn+1 are identity maps.

Remark 3.1.6. Similarity relation has reflexive property and symmetric one. If n = 1, by five Lemma, the middle arrow is an isomorphism, then it has the transitive property, too. If n ≥ 2, the transitivity could fail, hence we have to enlarge the relation.

Definition 3.1.7 (Equivalence relation). Let E, F ∈ YEn_A(A, B). We say that E and F are equivalent (E ∼ F ) if there exists a sequence G1_{, G}2_{, . . . , G}m

of extensions, such that E = G1_{, F = G}m _{and G}

i is similar to Gi+1 for all

i = 1, . . . , m − 1.

Remark 3.1.8. The Definition is well defined, indeed, allowing to concatenate morphisms, we get the transitive property. If n = 1, by five Lemma, the middle arrows are isomorphisms, then two extensions are similar if and only if they are equivalent.

The unpleasant part of this definition is that it is not easy to work with. We state the following proposition, which gives a characterization of the equivalence relation:

Proposition 3.1.9. Let E, F ∈YEn_A(A, B). The following conditions are equivalent:

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CHAPTER 3. YONEDA EXTENSIONS 32 2. There exists an extension G and a commutative diagram:

0 B E1 · · · En A 0

0 B G1 · · · Gn A 0

0 B F1 · · · Fn A 0.

id id

Proof. 1 ⇐ 2) It follows by the Definition.

1 ⇒ 2) We proceed by induction on n. If n = 1 the middle arrows are isomorphisms, then we can conclude reversing and composing maps. If n ≥ 2, then it is enough to show that we can construct such a diagram starting from one of the form:

0 B E1 · · · En A 0 0 B L1 · · · Ln A 0 0 B F1 · · · Fn A 0. id id id id Let E0

2 be the image of the map E1 → E2. We define the extension E2 to be:

0 −→ E0₂−→ E2 −→ · · · −→ En−→ A −→ 0.

Similarly we proceed for L2 _{and F}2_{. The diagram above induces the following}

one: 0 E₂0 E2 · · · En A 0 0 L0₂ L2 · · · Ln A 0 0 F₂0 F2 · · · Fn A 0. id id Let G0

2 be the direct sum E20 ⊕ F20. Let e : E20 → G02, f : F20 → G02 be

the canonical inclusions and let l : L0

2 → G02 be the canonical map. By

Proposition 1.2.5, we can define the extension E3 _{to be:}

0 −→ G0₂−→ G0₂a

E0 2

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CHAPTER 3. YONEDA EXTENSIONS 33 Similarly we proceed for L3_{and F}3_{. All these extensions are in YE}n−1

A (A, G02).

By universal property of pushout, there is the following commutative diagram: 0 G0₂ G0₂` E0₂E2 · · · En A 0 0 G0₂ G0₂` L0 2L2 · · · Ln A 0 0 G0₂ G0₂` F₂0F2 · · · Fn A 0. id id id id

By inductive hypothesis, there is 0 G0₂ G0₂` E02E2 · · · En A 0 0 G0₂ G0₂` L0 2L2 · · · Ln A 0 0 G0₂ G0₂` F20F2 · · · Fn A 0. id id id id

We define the exact sequence X to be: 0 → B → E1

a

B

F1→ G02 → 0,

where the first map is given by the inclusion on the first component and the second map is the canonical one. We can merge X on the left to the diagram above to get: 0 B E1`BF1 G02 ` E0 2E2 · · · En A 0 0 B E1`_BF1 G02 ` L0₂L2 · · · Ln A 0 0 B E1`BF1 G02 ` F0 2F2 · · · Fn A 0. id id id id id id

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CHAPTER 3. YONEDA EXTENSIONS 34 To conclude it is enough to construct two commutative diagrams:

0 B E1 E2 · · · En A 0 0 B E1 ` BF1 G02 ` E0₂E2 · · · En A 0 id id and 0 B F1 F2 · · · Fn A 0 0 B E1`BF1 G02 ` F0 2F2 · · · Fn A 0. id id

Let’s see the first first diagram, the construction of the second one is sim-ilar. We define E1 → E1`BF1 to be the canonical inclusion on the first

component and E2 → (E20 ⊕ F20)

`

E0₂E2 to be the canonical inclusion on the

second component. If we fix all the other maps to be the identity, then, by construction, the diagram commutes.

Remark 3.1.10. At the same way, it can be proved that two extensions E and F are equivalent if and only if there is a commutative diagram:

0 B E1 · · · En A 0 0 B L1 · · · Ln A 0 0 B F1 · · · Fn A 0. id id id id

Definition 3.1.11(Yoneda extensions set). Define the quotient YExtn_A(A, B) := YEn

A(A, B)/ ∼ to be the set of Yoneda extensions.

Define YExt0

A(A, B) to be Hom(A, B). We are going to construct an

operation · for all n, m non-negative integers YExtn

A(A, B) ×YExtmA(C, A) ·

−→YExtn+m_A (C, B) :

• (Composition) Suppose that n = m = 0, we define · to be the composi-tion map.

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CHAPTER 3. YONEDA EXTENSIONS 35 • (Pullback) Suppose that n > 0 and m = 0. Let [E] be a Yoneda extension and let f : C → A be a morphism. By Proposition 1.2.5 there is the following commutative exact diagram:

0 B E1 · · · En−1 En A 0

0 B E1 · · · En−1 En−1×AC C 0.

id id id f

Define the pullback [E] · f to be the class of the extension:

0 B E1 · · · En−1 En−1×AC C 0.

By universal property of fiber product, it is well defined.

• (Push-forward) Suppose that n = 0 an m > 0. Let g : A → B be a morphism and let [F ] be a Yoneda extension. By Proposition 1.2.5 there is the following commutative exact diagram:

0 B E1 E2 · · · En A 0

0 D D`

BE1 E2 · · · En A 0.

g _id _id id

Define the push-forward g · [F ] to be the class of the extension:

0 D D`

BE1 E2 · · · En A 0.

By universal property of pushout, it is well defined.

• Suppose that n, m > 0. Let [E] and [F ] be Yoneda extensions in YExtn

A(A, B) and YExtmA(C, A), respectively. Define [E] · [F ] to be the

class of the following extension:

0 → B → E1→ · · · → En→ F1 → · · · → Fm→ C → 0.

Proposition 3.1.12. Let [E] ∈ YExtn_A(A, B) and [F ] ∈ YExtm_A(C, A). If [E] or [F ] is the class of the trivial extension, then [E] · [F ] is the same class, too.

Sketch of the proof. If n > 0, m = 0 and f is the trivial map, then the pullback is given by

0 B E1 · · · En−1 En A 0

0 B E1 · · · En−1 En0 ⊕ C C 0,

Splitting families in Galois cohomology