Treball final de grau

GRAU DE MATEMÀTIQUES

Facultat de Matemàtiques i InformàticaUniversitat de Barcelona

Algebraic Groups and TannakianCategories

Autor: Guillem Sala Fernandez

Director: Dra. Teresa CrespoRealitzat a: Departament

De Matemàtiques i Informàtica

Barcelona, June 29, 2017

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Abstract

The main goal of this memoir is to introduce the notion of an algebraicgroup and study its properties, generalizing many common notions ingroup theory, such as representations and actions. In addition, we seethat there is a duality between affine algebraic groups and what we callHopf algebras. Afterwards, we see that we can define a category whoseobjects are finite representations of affine algebraic groups together withthe natural homomorphisms between them. This leads us to the necessityof introducing a more general structure for this kind of categories, whichwe call tannakian categories. Eventually, we apply the results we obtainwith these structures to differential Galois theory.

Resum

L’objectiu principal d’aquesta memòria és introduïr la noció de grup al-gebraic i estudiar les seves propietats, tot generalitzant nocions comuns ateoria de grups, com ara representacions i accions. A més, veiem que hiha una dualitat entre els grups afins algebraics i el que anomenem àlge-bres de Hopf. Despres d’això, veiem que podem definir una categoria queté per objectes les representacions finites de grups afins algebraics junta-ment amb els homomorfismes naturals entre aquestes. Això ens porta ala necessitat d’introduïr una estructura més general per a aquest tipus decategories, que anomenem categories tannakianes. Finalment, apliquemels resultats que obtenim amb aquestes estructures a la teoria diferencialde Galois.

Acknowledgements

Firstly, I would like to thank all the professors that have lectured methroughout the career for inspiring me, helping me learn and fulfil mydream of becoming a mathematician. I feel particularly thankful for allthe time professors Maria Jesús Carro, Artur Travesa and Teresa Crespohave dedicated to me during this last year. Each one of them has helpedme gain selfconfidence, which has been a major obstacle for me duringthe last couple years. I am sure these improvements will be reflected onmy work in a very positive manner.On the other hand, I want to thank my family, for supporting me from theday that I was born, for raising me in a loving enviroment, for providingme with everything I have needed and for showing me the importance ofhelping those in need. Also, let me thank my dad for showing me thebeauty of music, my mom for giving me this personality and my brotherfor inspiring me and being my best friend.Also, thanks to Albert, Bernat, and Marti. I treasure each and everysecond I have spent with them, they have inspired me in each possiblemanner. Hopefully someday I will dedicate them my PhD thesis, too.Finally, there is one person that deserves a whole paragraph, and that isVeronica. If only I could find the words to describe how thankful I amfor everything that she has done for me. I can only say that I want tothank her for her honesty, love and for the way she has treated me sincethe very first moment. This piece of work is for her. None of this wouldhave been possible without her.

Contents

Introduction 1

1 Schemes 31.1 Affine algebraic schemes . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 The Zariski topology . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Sheaves and (locally) ringed spaces . . . . . . . . . . . . . . . 7

1.2 Algebraic Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 Coherent sheaves and algebraic subschemes . . . . . . . . . . 14

1.3 Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Algebraic Groups 192.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Basic properties of algebraic groups . . . . . . . . . . . . . . . . . . . 212.3 Kernels and Group actions . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Affine Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4.1 The general linear group GLn . . . . . . . . . . . . . . . . . . 252.5 Homomorphisms and products . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 The Frobenius map . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Hopf Algebras 293.1 Bialgebras and affine monoids . . . . . . . . . . . . . . . . . . . . . . 293.2 Hopf Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Hopf subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Linear representations and characters of algebraic groups 354.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Properties of Γ(G,O)-comodules . . . . . . . . . . . . . . . . . . . . 384.3 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 Eigenspaces and the decomposition theorem . . . . . . . . . . 41

5 Tannakian Categories 435.1 Abelian Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Symmetric Monoidal Categories . . . . . . . . . . . . . . . . . . . . . 465.3 Internal homomorphisms and rigidity . . . . . . . . . . . . . . . . . . 485.4 Examples of tannakian categories . . . . . . . . . . . . . . . . . . . . 51

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6 Tannaka-Krein duality 556.1 The tannakian reconstruction . . . . . . . . . . . . . . . . . . . . . . 556.2 Tannakian Categories and Algebraic Groups . . . . . . . . . . . . . . 576.3 A note on fibre functors . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Application: Differential Galois Theory 637.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.1.1 LSX and ConnD . . . . . . . . . . . . . . . . . . . . . . . . . 637.1.2 DiffModC(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.2 Differential Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . 667.2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . 667.2.2 The differential Galois algebraic group . . . . . . . . . . . . . 68

Bibliography 73

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Introduction

All problems in mathematics arepsychological!

Pierre Deligne

The present text is a reflection of a series of interests that I have developed during myBSc in Mathematics. During the last two semesters, the theory of categories gainedmy attention because of its vast range of applications in mathematics, the way itrelates several topics in the field and allows to transfer results from one branch toanother in what I believe is a very beautiful manner.At the beginning of this year, Teresa Crespo talked to me about the theory of tan-nakian categories, which I immediately accepted as the subject of this thesis. At thebeginning, I knew few things about category theory so, in order to distance myselffrom the most basic definitions of this theory, I had to go through several commuta-tive algebra and algebraic geometry books, which allowed me to see what mathematicslooks like after this first stage of undergraduate level, and relate some topics that Iwas already familiar with to category theory. All in all, I am sure that the conceptsI have learnt will allow me to pursue not only a good MSc but also what I expect tobe a very interesting PhD thesis.During this memoir we go through several topics such as affine algebraic schemes,algebraic group theory and category theory. The combination of these three even-tually leads us to some very interesting results, such as the fundamental theoremof differential Galois theory using tools from category theory, showing how powerfulcategory theory is when mixed properly with different theories, such as the theory ofschemes that we mentioned before.Since there are different ways of approaching algebraic groups, which are a cornerstonein the development of this memoir, I considered that it is important to show the twomost common approaches, via theory of schemes and via category theory. Moreprecisely, an algebraic group can be introduced either by taking an algebraic varietytogether with a “multiplication” map and an identity and inverse map such that thelatter make a couple of diagrams commute or by taking a functor from the category ofalgebras over an algebraically closed field into the category of groups and asking thecomposition of the previous with the forgetful functor that forgets the group structureinto the category of sets to be representable (in the categorical sense) by some algebraover the field. In Chapter 3 we see these definitions with more detail and we see that,in turn, they are both equivalent.

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For this reason, Chapter 1 is entirely dedicated to a short yet precise introduction tothe theory of Schemes together with some “flashbacks” of category theory. Further-more, in this same chapter, we see some of the results that come in handy during thememoir and that allow us to develop the rest of the theory.In chapter 2 we give an introduction to algebraic group theory, giving some basicresults and examples that lead us to the construction of some more complex algebraicgroups such as the Frobenius map, as an application of how to use the link betweenalgebraic groups and algebraic schemes.On the other hand, we introduce in Chapter 3 the notion of Hopf algebras, whichare the most common way of dealing with affine algebraic groups, thanks to theduality that exists between them. It is worth mentioning that Hopf Algebras are notonly of a big interest in the development of algebraic group theory, but also on thedevelopement of other branches of science such as physics, although we do not deepenmuch into this latter point. Hence, Hopf algebra theory forms an important branchinside not only mathematics but physics and has a wide variety of applications.Linear representations are just the tip of the iceberg, and they lead us to the proofof the fundamental theorem of differential Galois theory, so that is why we give ashort introduction to them in Chapter 4. Along the chapter, we show the relation-ship between actions and representation theory which follows naturally in the case ofstandard group theory. We introduce as well the notion of comodules, which gives abijective correspondence between linear representation of algebraic groups over vectorspaces over a field and the comodule structures on these vector spaces. As an appli-cation, we prove the Theorem of decomposition of a representation into characters.By the time we arrive to Chapter 4, the reader should notice that we no longeruse the scheme theoretic definition of an algebraic group but instead, we work allthe time with the categorical one. For this same reason, in Chapter 5 we introduceTannakian categories, which is the last step towards the main theorem of this book,the Tannaka-Krein duality theorem, that we introduce in Chapter 6. These last twochapters show the reader the importance of Tannakian Categories, following a veryconstructive introduction to these, starting with abelian categories and ending upwith the definition of a fiber functor, which leads to the definition of a Tannakiancategory, along with some important examples. To conclude the introduction to thisabstract notion, we show the previously mentioned Tannaka-Krein duality theorem,which states that each tannakian category is equivalent to the category of finitedimensional representations of a certain algebraic group.This last theorem is the key to the proof of the fundamental theorem of differentialGalois theory, so in order to conclude the memoir, we apply all that we have learntin Chapter 7, giving a short introduction to differential algebra and showing therepeatedly mentioned theorem.In short, this memoir represents a brief introduction to the theory of algebraic groupsand tannakian categories. It goes through several branches of mathematics in orderto inspire the reader to apply these theories, which are quite beautiful. I have tried todevelop the theory in a way that this is a self-contained memoir and also, in order tointroduce this theory to students who are currently finishing a BSc in Mathematics.

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Chapter 1

Schemes

1.1 Affine algebraic schemesThroughout this work, k will denote a field and A a finitely generated k-algebra. Infact, we will simply refer to finitely generated k-algebras as k-algebras. Also, we willdenote the category of finitely generated k-algebras as Algk.

1.1.1 The Zariski topologyDefinition 1.1.1.1. Let X be the set of maximal ideals in A and a an ideal in A.We will call the set

Z(a) = m ∈ X : a ⊂ m

zero set of a.

It is important to notice that if our k-algebra A is k[X1, . . . , Xn], this definitioncoincides with the definition of an algebraic set V (S) from algebraic geometry, thatis, the set of common zeroes of a family of polynomials S in A, because if we let abe the ideal generated by a set of polynomials in A, S = gii∈I , then any elementx ∈ a can be written in the form

x =∑i∈I

figi,

where each fi ∈ k[X1, . . . , Xn] and therefore, such sum is zero at every point at whichthe gi are all zero. Thus, V (S) ⊆ V (a). The reverse inclusion is clear, hence thealgebraic subsets of kn are the zero sets of ideals in k[X1, . . . , Xn].Also, according to the following proposition, we can endow X with a certain topology.

Proposition 1.1.1.2. Let a and b be a pair of ideals and aii∈I a family of ideals.Then,

1. Z(0) = X,

2. Z(A) = ∅,

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3. Z(ab) = Z(a ∩ b) = Z(a) ∪ Z(b) and

4. Z(∑

i∈I ai) = ∩i∈IZ(ai).

Proof. All the results follow straightforwardly from the definition.

Clearly, this shows that the sets Z(a) are the closed sets for a topology on X, whichis called the Zariski Topology on X. For every x ∈ X, we will denote mx the point xseen as a maximal ideal. Notice that we can define the residue field at point x ∈ X asκ(x) := A/mx. Also, we will denote the set X together with this topology as spm(A)and An := spm(k[X1, . . . , Xn]).Now, let S be a subset of spm(A) and let I(S) denote the intersection of all the idealsof S, namely

I(S) :=∩m∈S

m.

We refer to I(S) as the ideal associated to S. Also, let us remember the definition ofthe radical of an ideal a,

rad(a) := x ∈ A : xr ∈ a, for some r ∈ N,

and recall that every prime ideal is also a radical ideal (this can be shown by inductionover r ∈ N, if x ∈ A, then xr ∈ a implies x ∈ a). Let us remember as well the followingresult from algebraic geometry (see Theorem 4.3 of [Per]).

Theorem 1.1.1.3. (Hilbert’s Strong Nullstellensatz). For every ideal a in A,

I(Z(a)) :=∩

m∈Z(a)

m =∩

m∈X:a⊂m

m = rad(a).

Clearly, if a is a radical ideal, I(Z(a)) = a.

Notice that every closed subset of X can be written as Z(a) for some ideal a inA. Therefore, via Hilbert’s Nullstellensatz, we have that Z and I define a bijectivecorrespondence between the set of closed sets of spm(A) and the set of radical idealsof A. In fact, since every prime ideal p satisfies rad(p) = p, we have that prime idealscorrespond to irreducible sets of spm(A).Now that we have studied the closed sets of the Zariski topology, we can move on tothe study of the open sets of that same topology. For each f ∈ A, let D(f) := m ∈X : f /∈ m. Clearly,

D(f) := m ∈ X : f /∈ m = X \ m ∈ X : f ∈ m = X \ Z(⟨f⟩)

that is, D(f) is an open set because it is the complementary of a closed set. Wewill call these sets basic open subsets of spm(A). On the other hand, Atiyah andMacdonald show in Theorem 7.5 of [Ati] the following, well-known theorem.

Theorem 1.1.1.4 (Hilbert’s Basis Theorem). If k is a Noetherian ring, then the ringk[X1, . . . , Xn] is also a Noetherian ring.

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And thanks to this theorem, we can show the following proposition, because eachk-algebra is Noetherian since all fields are noetherian and a quotient of a noetherianring is noetherian (if R/I is a quotient of a noetherian ring, any ideal J ⊂ R/I is ofthe form J/I for some ideal I ⊂ J ⊂ R).

Proposition 1.1.1.5. The basic open subsets form a basis for spm(A).

Proof. Acording to Hilbert’s Basis Theorem, A is noetherian, because it is a finitelygenerated k-algebra. Therefore, a is finitely generated, thus there exist f1, . . . , fm ∈ Asuch that a = ⟨f1, . . . , fm⟩ and

X \ Z(a) = m ∈ X : a ⊂ m =∪

1≤i≤m

m ∈ X : fi /∈ m =∪

1≤i≤m

D(fi),

so, since any open set of spm(A) can be written as the union of basic open subsets,the set of basic open subsets forms a basis of spm(A).

It is also interesting to see the following proposition, which gives us some usefulproperties of the basic open subsets.

Proposition 1.1.1.6. Let f, g be elements of a k-algebra A, and let D(fi)i∈I be afamily of basic open subsets of spm(A).

1. D(fi)i∈I forms an open covering of spm(A) if and only if 1 can be written as

1 =∑i∈I

aifi,

where ai ∈ A for each i ∈ I, with only a finite number of non-zero terms. Infact, spm(A) is quasi-compact.

2. D(f) ∩D(g) = D(fg).

3. Given f, g ∈ A, D(g) ⊆ D(f) if and only if g ∈ rad(⟨f⟩).

Proof. Since

X \ Z

(∑i∈I

⟨fi⟩

)=∪i∈I

D(fi),

we have that the open sets form a covering if and only if the ideal generated by allfi is the whole ring A, namely 1 ∈

∑i∈I⟨fi⟩. In fact, since A is a finitely generated

k-algebra, this shows that any covering by basic open sets can be reduced to a finiteone and, since these basic open subsets of X form a basis for the Zariski topology,any open cover can be refined to one such that its sets are all basic, thus allowing tofind a finite covering. This shows the first point.In order to show the second point, all we have to notice is that, given m ∈ X, f ∈ mand g ∈ m if and only if fg ∈ m, because m is a maximal ideal.Finally, remember that the fact that A is a finitely generated k-algebra implies thatrad(a) = ∩a⊆mm, because for a general ring, both Jacobson radical and nilradical are

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equal if the ring is a finitely generated k-algebra. Therefore, Z(b) ⊆ Z(a) impliesthat

rad(a) =∩

m∈Z(a)

m ⊆∩

m∈Z(b)

m = rad(b),

and if m ∈ Z(b) and rad(a) ⊆ rad(b), then a ⊆ rad(a) ⊆ rad(b) ⊆ m impliesm ∈ Z(a), therefore we have that Z(b) ⊆ Z(a) if and only if rad(a) ⊆ rad(b) and, inparticular, that Z(a) = Z(rad(a)), so in order to show the third point, we have thatD(g) ⊆ D(f) is true if and only if Z(⟨f⟩) ⊆ Z(⟨g⟩), and by the previous observation,this is true if and only if ⟨g⟩ ⊆ rad(⟨f⟩), that is, if and only if g ∈ rad(⟨f⟩).

Let us review now the relationship between Algk and the category of topologicalspaces, Top. In order to do so, we must remember a result from algebraic geometry.

Theorem 1.1.1.7 (Zariski’s Lemma). If B is a finitely generated algebra over a fieldk and B is a field, then B is a finite field extension of k.

Proof. See [Ati], Exercise 18 of Chapter 5.

Proposition 1.1.1.8. The morphism of categories spm : Algk → Top given byA 7→ spm(A) is a contravariant functor.

Proof. We have to see that for every A ∈ Ob(Algk), spm(A) ∈ Ob(Top), and thatfor every α ∈ HomAlgk

(A,B), spm(α) =: α∗ ∈ HomTop (spm(B), spm(A)).The first point is clear by construction, because we can endow each A ∈ Ob(Algk)with the Zariski topology, obtaining spm(A) ∈ Ob(Algk). Secondly, let m be amaximal ideal in B. Since B is a finitely generated k-algebra, so is B/m. Therefore,since m is a maximal ideal, B/m is a field and, according to Zariski’s Lemma, itis a finite field extension of k. Furthermore, the image of A in B/m is an integraldomain of finite dimension over k, thus it is a field, and that image is isomorphic toA/α−1(m), so α−1(m) is a maximal ideal in A. We only have to see that the map

α∗ : spm(B)→ spm(A)

that assigns to each m in B α−1(m) is continous, but that is clear, because for everyf ∈ A,

(α∗)−1(D(f)) = n ⊆ B : α∗(n) ∈ D(f) = n ⊆ B : α−1(n) ∈ D(f)

= n ⊆ B : f ∈ α−1(n) = n ⊆ B : α(f) ∈ n = D(α(f)),

that is, the preimage of an open set is an open set. Thus, spm is a contravariantfunctor.

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1.1.2 Sheaves and (locally) ringed spacesIn order to give a precise definition of the notion of sheaf, we will first give thedefinition of a sheaf of rings, and then show that the definition can be extended tothe definition of sheaf of k-algebras, that is, the notion can be easily translated fromrings to k-algebras and, in fact, to many other categories.

Definition 1.1.2.1. Let X be a topological space. A presheaf O of rings on Xconsists of the data

1. For every open set U ⊂ X, a ring O(U). The elements of O(U) are calledsections over U . Usually, we denote O(U) as Γ(U,O).

2. O(∅) is the trivial ring.

3. For each pair of open sets U and V of X, such that U ⊆ V , a homomorphismof rings ρVU : O(V )→ O(U) called restriction map with the conditions

(a) ρUU is the identity map.(b) For each triple of open subsets U ⊆ V ⊆ W of X, ρVU ρWV = ρWU .

Furthermore, given a presheaf O on a topological space X, we will say that O is asheaf if it satisfies the sheaf axiom, that is,

4. if U = ∪i∈IUi is an open covering of an open set U and fii∈I is a set ofelements fi ∈ O(Ui) for all i ∈ I such that fi

∣∣Ui∩Uj

= fj∣∣Ui∩Uj

, then there existsa unique f ∈ O(U) such that f

∣∣Ui

= fi for all i ∈ I.

A sheaf of k-algebras is a sheaf of rings such that moreover O(U) is a k-algebra forevery U and ρVU is a k-algebra homomorphism for each pair U, V .

Example 1.1.2.2. One of the most common examples that we can observe is thecase of the category of manifolds of differentiability class Cm, which will be denotedas Manm. Let M ∈ Ob(Manm). For each open set U of M , if we let Fm(U) denotethe set of real-valued Cm functions on U , we have that under point-wise addition andmultiplication, Fm(U) is a ring. Also, if V ⊆ U are open subsets of M , the restrictionhomomorphism is

ρVU : Fm(U)→ Fm(V ),

given by the restriction of functions. The other points of the definition are easily seenand, therefore, Fm is a sheaf of rings on M .

It is also interesting to notice that smooth manifolds can be defined in another way,that is, instead of being defined as topological spaces with a certain open cover satis-fying a series of conditions, they can be defined as a topological space together witha sheaf satisfying a certain property.

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Definition 1.1.2.3. Let X be a topological space and O a sheaf of rings on X. Wecall stalk of O at x

Ox := lim−→x∈U⊆XUopen

O(U) =

⨿x∈U⊆XUopen

O(U)

/ ∼,where for f ∈ O(V ) and g ∈ O(U), f ∼ g if and only if there exists W ⊆ U ∩ V suchthat f |W = g|W .

Definition 1.1.2.4. Let O and O′ be two sheaves on a topological space X. Amorphism of sheaves φ : O → O′ is a collection of maps φ(U) : O(U)→ O′(U) U⊆X

Uopensuch that for every V open in X with U ⊂ V , the following diagram commutes

O(V ) O′(V )

O(U) O′(U)

φ(V )

ρVU (ρ′)VU

φ(U)

Also, it is important to see how to create a sheaf from an existing one, for that reason,we give the next definition.

Definition 1.1.2.5. 1. Let O be a sheaf on a space X and let U be an open subsetof X. We can define the restriction sheaf O|U on U by taking OU(V ) := O(V ),for any open subset V of U . Clearly, the restriction sheaf is, indeed, a sheaf.

2. Let X and Y be topological spaces, let OX be a sheaf on X and also let f :X → Y be a continuous function. We can define the pushforward sheaf f∗OXon Y as the sheaf given by

f∗OX(U) := OX(f−1(U)),

for any open set U in Y , together with the necessary restriction maps. Thepushforward sheaf is also a sheaf.

Let us take a moment to study the ring of fractions of A, S−1A, where S is a multi-plicative set. Firstly, we have that if Sf := 1, f, f 2, . . . , clearly

Af := S−1f A ≃ A[T ]

⟨1− fT ⟩≃ A

[1

f

].

On the other hand, if D is an open subset of X, we can define

SD := A \∪m∈D

m,

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which is a multiplicative set because if st ∈ SD, then there exists m ∈ D such thatst ∈ m. That implies s ∈ m or t ∈ m, which is equivalent to s ∈ SD or t ∈ SD. Infact, the inclusion Sf ⊂ SD(f) gives a map

S−1f A→ S−1

D(f)A

that is an isomorphism. This will be of great use when we endow spm(A) togetherwith a sheaf of k-algebras. Finally, if D′ ⊂ D, SD′ ⊃ SD, thus there exists a canonicalmap S−1

D A → S−1D′ A. Hence, let us see how can we apply all these observations to

spm(A). All we have to do is show the following proposition.

Proposition 1.1.2.6. There exists a unique sheaf OX of k-algebras on X = spm(A)such that for every basic open subset D of X, OX(D) = S−1

D A. We will refer tospm(A) together with that sheaf of k-algebras as Spm(A). Also, for any x ∈ X, wehave that OX,x = Amx and κX(x) = Amx/mxAmx.

Proof. All we have to do is check that the OX given by OX(D) := S−1D A is a sheaf

of k-algebras. The first two points of the definition can be verified easily, becausethe ring of fractions of a k-algebra is a ring. For the third point, we know from theobservation preceding this proposition that for two given open subsets D ⊂ D′ of X,there exists a canonical homomorphism of rings

ρD′

D : S−1D A→ S−1

D′ A

and from that follows the transitivity. The fourth point is clear as well. The pointregarding the stalks and residue fields follows from the existence of that sheaf ofk-algebras, because for each open subset, D = D(f), S−1

D A ≃ Af , hence

OX,x := lim−→x∈U⊆XUopen

OX(U) = lim−→x∈U⊆XUopen

S−1U A = Ax,

and, thanks to Corollary 7.10 of [Ati], we have that κX(x) = Amx/mxAmx .

Notice that the sheaf from 1.1.2.6 has a very good property, and that is that its stalksat each point x ∈ X are local rings. We refer to it as the structure sheaf.

Definition 1.1.2.7. A k-ringed space is a pair (X,OX), where X is a topologicalspace and OX is a sheaf of k-algebras on X. A morphism between k-ringed spacesϕ : (X,OX) → (Y,OY ) is a pair (ϕ∗, ϕ♯) such that ϕ∗ : X → Y is a continous mapand ϕ♯ : OY → ϕ∗

∗OX is a morphism of sheaves.

So we can finally give the definition of affine algebraic scheme over a field k.

Definition 1.1.2.8. An affine algebraic scheme over k is a k-ringed space (X,OX)isomorphic to Spm(A) for some k-algebra A, in the sense that there exists a bijectivemorphism between both k-ringed spaces. We call morphism (or regular map) of affinealgebraic schemes over k any morphism of k-ringed spaces.

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Now, let us remember that given two categories C,D and two functors F and G inFunc(C,D), a natural transformation η or morphism from F to G is a family ofmorphisms that satisfy

1. ∀X ∈ Ob(C), there exists a morphism ηX : F (X) → G(X) between objects ofD, which is called component of η at X and

2. For each f ∈ HomC(X,Y ), the component morphisms make the following dia-gram commute

F (X) F (Y )

G(X) G(Y ).

F (f)

ηX ηY

G(f)

In case F and G are contravariant, the horizontal arrows in the previous diagram arereversed, and we say that F and G are (naturally) isomorphic if there exists a naturalisomorphism from F to G, that is ∀X ∈ Ob(C), ηX is an isomorphism in D. We willdenote the set of natural transformations from F to G as Nat(F,G).Remember as well that F : C → D is a (contravariant) equivalence of categories ifthere exists a (contravariant) functor G : D→ C such that FG and GF are naturallyisomorphic to the identity functors idD and idC respectively. G is called a quasi-inverseof F .The following result will allow us to develop the upcoming theory, as it will give usthe relationship between affine algebraic schemes and k-algebras. In turn, we see inChapter 3 that if we endow affine algebraic schemes with the structure of a “group”,we have a strong relation between such structures and Hopf algebras.

Proposition 1.1.2.9. The morphism Spm : Algk → AffSchk, is a contravariantequivalence from Algk to the category of affine algebraic schemes over k, AffSchkwith quasi-inverse AffSchk → Algk given by

(X,OX) 7→ OX(X).

In fact, we have that, for any A,B ∈ Ob (Algk),

HomAlgk(A,B) ≃ HomAffSchk

(Spm(B), Spm(A)).

Proof. First, we must see that Spm is a contravariant functor. Clearly, given A ∈Ob(Algk) and considering the definition of Spm(A), we have Spm(A) ∈ Ob(AffSchk).Secondly, we know that given α ∈ HomAlgk

(A,B), there exists a continous morphismof topological spaces α∗ ∈ Hom(spm(B), spm(A)) (cf. prop. 1.1.4). Our goal is tofind a regular map of affine schemes α : Spm(B)→ Spm(A) such that its underlyingtopological homomorphism is |α| = α∗. We showed in prop. 1.1.5 that given a k-algebra C, there exists a unique sheaf such that for any basic open set D of spm(C),

10

Γ(U,Ospm(C)) = S−1D C. Therefore, in order to define α, it suffices to do it for the sec-

tions over basic open sets, because it verifies the sheaf axiom and therefore by a sim-ple patching argument the result follows. Remember that (α∗)−1(D(f)) = D(α(f)),hence we have that

Γ(D(f), α∗∗Ospm(B)) = Γ((α∗)−1D(f),Ospm(B)) = Γ(D(α(f)),Ospm(B)) = S−1

D(α(f))B

thus the morphism α : A→ B induces the morphism

α♯(D(f)) : Γ(D(f),OSpm(A)) = S−1f A→ S−1

D(α(f))B = Γ(D(f), α∗∗OSpm(B)),

thus we have definedα♯ : Ospm(A) → α∗

∗Ospm(B)

Now, since Γ(spm(A),Ospm(A)) = A, we have that if we take α ∈ HomAlgk(A,B),

there is the induced map α♯ : Ospm(A) → α∗∗Ospm(B) that, by taking sections, satisfies

A = Γ(spm(A),Ospm(A))→ Γ(spm(A), α∗∗Ospm(B)) = B

which is the same as α, therefore we have it.

Definition 1.1.2.10. Given an affine scheme X over k and A a k-algebra,

X(A) := HomAffSchk(Spm(A), X),

and any homomorphism of X(A) is called an A-point.

1.2 Algebraic SchemesLet (X,OX) be a k-ringed space. An open subset U of X is said to be affine if(U,OX |U) is an affine algebraic scheme over k. We sometimes refer to (U,OX |U) asan open subscheme as well. Hence, let us give the following definition.

Definition 1.2.0.1. An algebraic scheme over k or algebraic k-scheme is a k-ringedspace (X,OX) that admits a finite covering by affine open subsets. A morphism ofalgebraic schemes or regular map over k, is a morphism of k-ringed spaces. If (X,OX)is an algebraic k-scheme, Γ(X,OX) is called the coordinate ring of X. Also, an opensubscheme of X is a pair (U,OX |U). We will denote the category of algebraic schemesas AlgSchk.

From now on, whenever there is no possible confusion with the notation, we willdenote X the algebraic scheme (X,OX), and |X| the underlying topological space ofX. Furthermore, if the base field k is known, we will simply write algebraic schemeinstead of algebraic scheme over k.Also we say that a regular map Y → X of algebraic schemes is surjective, injective,open and/or closed whenever the underlying map of topological spaces |Y | → |X| issurjective, injective, open and/or closed respectively.

11

Proposition 1.2.0.2. Let X be an algebraic scheme over k. Then |X| is a noetheriantopological space.

Proof. In the affine case, since A is a finitely generated k-algebra, spm(A) is noethe-rian. For the general case, let us write X = ∪ni=1Ui, and let Vj ⊆ Vj+1 be an ascendingchain of open subsets of X. Clearly, Vj = ∪ni=1(Ui ∩ Vj), and so the Ui ∩ Vj forms anascendent chain of open subsets of Ui, which is noetherian, and therefore, the chaingiven by the Vj is stationary.

Remark 1.2.0.3. Let K be a field containing the base field k. Clearly, there exists afunctor from AlgSchk → AlgSchK that sends any algebraic k-scheme to an algebraicK-scheme. In fact, if X = Spm(A) for a certain k-algebra A, XK = Spm(K ⊗k A),because the tensorial product turns A into a K-algebra K ⊗k A if we also endow itwith the product by elements of K in the first component.

Notice that given X an algebraic scheme over k and A a k-algebra, Proposition 1.1.2.9gives us an isomorphism

HomAlgSchk(X, Spm(A)) ≃ HomAlgk

(A,OX(X)).

Therefore, it would be interesting to study some of the functorial properties of alge-braic schemes. In order to do so, let hA denote the functor

hA : Algk → Set

R 7−−−→ HomAlgk(A,R).

We will say that a functor F ∈ Func(Algk,Set) is representable if it is isomorphicto hA for some k-algebra A.We say that for a ∈ F (A), the pair (A, a) represents F if the natural transformation

Ta : hA → F

f 7−→ F (f)(a).

is an isomorphism. In that case, we will say that a is universal.Recall now that given a functor F : C → D, where C and D are categories, if wedefine for each X,Y ∈ Ob(C) a function

FX,Y : HomC(X,Y )→ HomD(F (X), F (Y )),

we will say that the functor F is

1. Faithful if FX,Y is injective,

2. Full if FX,Y is surjective,

3. Fully faithful if FX,Y is bijective for all X,Y ∈ Ob(C) that is, if it is full andfaithful.

12

for each X,Y ∈ Ob(C).Now let us remember Yoneda’s lemma. Its proof can be found in [Ma], pages 59-62.

Theorem 1.2.0.4 (Yoneda’s Lemma). If C is a category, then the functor

h : C→ Func(C,Set)

A 7−→ hA.

where hA(S) := HomC(A, S) is fully faithful, so

HomC(A,B) ≃ HomFunc(C,Set)(hB, hA).

In fact, if we let F : C→ Set be a functor and X ∈ Ob(C), there exists a bijection

HomFunc(C,Set)(hX , F )

∼←−→ F (X)

given by α 7→ αX(idX).

This means that for each B ∈ Ob(Algk), and each functor F : Algk → Set, thereexists x ∈ F (B) that defines an homomorphism Hom(B,R) → F (R), sending f ∈Hom(B,R) to F (f)(x). Also, this homomorphism is natural in R, so there is a mapof sets

F (B)→ Nat(hB, F ).

Eventually, this is a bijection. Thus, for F := hA,

Hom(A,B) ≃ Nat(hB, hA),

so the contravariant functor that sends A to hA is fully faithful. Let now hX be thefunctor hX := HomAlgSchk

(−, X) from algebraic schemes over k to sets. Yoneda’slemma states that given two algebraic schemes X and Y ,

Hom(X,Y ) ≃ Nat(hX , hY ).

If we let haffX denote the functor

haffX : Algk → Set

A 7−→ X(A),

we have that haffX = hX Spm, and therefore we can regard it as the restriction of

hX to affine algebraic schemes. Also, since every natural transformation haffX → haff

Y

extends uniquely to a natural transformation hX → hY , we have that

Nat(haffX , h

affY ) ≃ Nat(hX , hY ) ≃ Hom(X,Y ).

The latter isomorphisms show that the functor

F : AlgSchk → Func(Algk,Set)

X 7−−−−−−→ haffX

is fully faithful, because for each X,Y ∈ Ob(AlgSchk), FX,Y is an isomorphism.

13

Remark 1.2.0.5. Finally, fix a family (Ti)i∈N, and let Alg0k denote the full subcat-

egory of Algk of objects of the form k[T0, . . . , Tn]/a for some n ∈ N and an ideal ain k[T0, . . . , Tn]. The inclusion Alg0

k → Algk is an equivalence of categories, but theobjects of Alg0

k form a set, and so the set-valued functors on Alg0k form a category.

We will call the objects of Alg0k small k-algebras. Also, we let X be the functor

Alg0k → Set defined by an algebraic scheme. Thanks to Yoneda’s lemma, the functor

that sends X 7→ X is fully faithful. Notice that if a functor F : Alg0k → Set is rep-

resentable by an algebraic scheme X, then X is uniquely determined up to a uniqueisomorphism, and X extends F to a functor from Algk to Set.

To finish this section, it would be interesting to see an example of an algebraic schemethat is not affine.

Example 1.2.0.6. Let X = A2k \ 0. On the one hand, notice that since A2

k =Spm(k[X1, X2]), and thanks to 1.1.2.9 we have that the restriction map

Γ(A2k,OA2

k) = k[X1, X2]→ Γ(X,OX)

is bijective. If we take an element of Γ(X,OX), it is represented by a pair (f, g) ∈OX(D(X1))×OX(D(X2)) = k[X1, X2]X1 × k[X1, X2]X2 , where f and g coincide overD(X1) ∩D(X2) = D(X1X2). By a simple calculation, the latter implies that f = gbelongs to k[X1, X2]. Hence, Γ(X,OX) = k[X1, X2]. Therefore, if X was an affinealgebraic scheme, there would be a canonical isomorphism

X → Spm(Γ(X,OX)) ≃ Spm(k[X1, X2]) = A2k,

which is false because 0 /∈ X.

1.2.1 Coherent sheaves and algebraic subschemesDefinition 1.2.1.1. Let (X,OX) be a ringed space. A sheaf of OX-modules or OX-module is a sheaf F on X such that

1. For each open set U of X, the abelian group F(U) is an OX(U)-module and

2. For each inclusion morphism of open sets V ⊆ U , the restriction morphismF(U) → F(V ) is compatible with the module structures via the ring homo-morphism ρVU : OX(U)→ OX(V ).

Also, a morphism F → G of OX-modules is a morphism of sheaves such that for eachopen set U of X, the map F(U)→ G(U) is a homomorphism of OX(U)-modules. IfF(U) is an ideal in OX(U) for all open sets U ⊂ X, we say that F is a sheaf of idealson X.

One of the most common examples of OX-modules is the following. Take a morphismof schemes ϕ : X → Y , and define

J (U) := ker(OY (U)→ ϕ∗OX(U)),

for each open subset U of X. Clearly, J is a sheaf of ideals.

14

Proposition 1.2.1.2. Let X = Spm(A) be an affine scheme and let M be an A-module. There is a unique OX-module M satisfying

1. M(D(f)) =M ⊗A Af ,∀f ∈ A.

2. For each x ∈ X, the stalk Mmx is isomorphic to the localized module Mmx.

Proof. It can be found in [Har], Proposition 5.1 of Chapter II.

Definition 1.2.1.3. Let X be an algebraic scheme. An OX-module F on X is aquasi-coherent sheaf if there is an open affine cover Ui = Spm(Ai)i∈I of X suchthat for each i ∈ I the restriction F|Ui

is isomorphic to an OUi-module of the form

Mi, where Mi is an Ai-module. If the previous conditions are given and each Mi isa finitely generated Ai-module, we say that F is a coherent sheaf. Furthermore, F islocally free if the Mi are free Ai-modules.

It is worth mentioning that in Corollary 5.5, Chapter II of [Har], Hartshorne showsthat in the affine case, the functor M 7→ M gives an equivalence of categories betweenthe category of A-modules, ModA, and the category of quasi-coherent OX-modules,that is commonly denoted as QCoh(OX). Finally we can give the definition of asubscheme.

Definition 1.2.1.4. Let X be an algebraic k-scheme and let J be a coherent sheafof ideals in OX . Let Z be the support of the sheaf OX/J . We call the algebraicscheme (Z, (OX/J )|Z) closed subscheme of X defined by J . Also, a subscheme of analgebraic scheme X is a closed subscheme of an open subscheme of X.

Hartshorne shows in Proposition 5.9 of Chapter II in [Har] that Z is a closed subsetof X and in Corollary 5.10 that Z ∩ U is affine for every open affine U of X.

1.3 Algebraic VarietiesFirst let us give sense to the concept of product of schemes. In order to do so, wemust remember first the following definition from category theory.

Definition 1.3.0.1. Let X,Y, Z ∈ Ob(C), ϕ ∈ HomC(X,Y ) and ψ ∈ HomC(Z, Y ).A fibre product of ϕ and ψ is an object X ×Y Z ∈ Ob(C) together with morphismsp ∈ HomC(X ×Y Z,X) and q ∈ HomC(X ×Y Z,Z) called projection morphisms suchthat the following diagram

X ×Y Z q//

p

Z

ψ

Xϕ // Y

commutes, and such that the following universal property holds: for any W ∈ Ob(C)and morphisms α ∈ HomC(W,X) and β ∈ HomC(W,Z) with ϕ α = ψ β there is a

15

unique γ ∈ HomC(W,X ×Y Z) making the diagram

Wβ

++VVVVVVVVV

VVVVVVVVVV

VVVVVVVV

γ ((QQQQQQQ

α

!!BBB

BBBB

BBBB

BBBB

BBBB

B

X ×Y Zp

q// Z

ψ

Xϕ // Y

commute. Also, we say that a commutative diagram

W //

Z

X // Y

in a category C is cartesian if W and the homomorphisms W → X and W → Z forma fibre product of the morphisms X → Y and Z → Y .

Theorem 1.3.0.2. Fibre products exist in the category of algebraic k-schemes, andthey are unique up to isomorphism.

Proof. Let X,Y and S be algebraic k-schemes and let X → S and Y → S be twogiven morphisms. All we have to do is prove it for affine algebraic k-schemes andthe result will follow using a gluing argument. Let A,B and R be k-algebras suchthat X = Spm(A), Y = Spm(B) and S = Spm(R). Our goal is to show thatX ×S Y = Spm(A⊗R B). We know that for any scheme Z,

HomAlgSchk(Z, Spm(A⊗R B)) ≃ HomAlgk

(A⊗R B,Γ(Z,OZ)),

and also that for any ring T , giving a homomorphism A ⊗R B → T is the same asgiving two homomorphisms A→ T and B → T that induce the same homomorphismon R. If we let T = Γ(Z,OZ), thanks to the previous fact, we have that givinga homomorphism of schemes Z → Spm(A ⊗R B) is the same as giving morphismsZ → X and Z → Y that give rise to the same morphism of Z into S. Finally,from the definition of fibred product follows that the previous scheme is unique upto isomorphism, if it exists. The rest is easy to see by following a simple patchingargument, glueing sheaves, schemes and morphisms of schemes.

In other words, given any solid commutative diagram of morphisms of schemes

Z

++VVVVVVVV

VVVVVVVV

VVVVVVVV

VV

((PPPPPPP

BBB

BBBB

BBBB

BBBB

BBBB

X ×S Y

// Y

X // S

16

there exists a unique dotted arrow making the diagram commute. In particular, noticethat this means that if X,Y and S are algebraic k-schemes, then X ×S Y is also analgebraic scheme, and the same happens for affine algebraic k-schemes.It is important to notice that when ϕ and ψ are the structure maps, that is, whenϕ : X → Spm(k) and ψ : Y → Spm(k), the fibre product becomes the product, whichwe will denote as X × Y , and

HomAlgSchk(T,X × Y ) ≃ HomAlgSchk

(T,X)× HomAlgSchk(T, Y ).

Definition 1.3.0.3. Let X ∈ Ob(AlgSchk). The regular map ∆X : X → X × Xsuch that its composites with the projection maps equal the identity map of X iscalled the diagonal map of X. We say that X is separated if ∆X(X) is closed inX ×X.

Proposition 1.3.0.4. Affine schemes over algebraically closed fields are separated

Proof. It can be found in [Per], Proposition 2.8 of Chapter VII.

On the other hand, remember that a ring A is reduced if it has no nonzero nilpotentelements. We say that an algebraic scheme X is reduced if the local ring OX,x isreduced for all x ∈ X. It is crucial to notice that given a k-algebra A, Spm(A) isreduced if and only if A is reduced.In addition to this, we say that an algebraic scheme X is geometrically reduced if Xover k is reduced. Throughout this work, we will essentialy focus on the affine case,so we simply mention some of the results that concern us the most in the followingproposition.

Proposition 1.3.0.5. Let A be a k-algebra and let X be an algebraic scheme over k.

1. Spm(A) is geometrically reduced if and only if A is an affine k-algebra, that is,if k ⊗k A is reduced.

2. If X is geometrically reduced, for every field K containing k, X is reduced overK.

3. If X is geometrically reduced and Y is reduced (resp. geometrically reduced),then X × Y is reduced (resp. geometrically reduced).

Notice that since we are supposing that k = k most of the time, in the first point ofthe previous proposition, we have k ⊗k A ≃ A, so we simply have to check wether Ais reduced or not. Let us give a final definition.

Definition 1.3.0.6. An algebraic variety over k is an algebraic scheme over k thatis both separated and geometrically reduced.

And so we have that affine schemes over reduced k-algebras are algebraic varieties,so we can refer to them either as affine schemes or as affine algebraic varieties.

17

18

Chapter 2

Algebraic Groups

2.1 DefinitionThroughout this section, k will denote a fixed field, not necessarily algebraicallyclosed. Let us start by giving the definition of algebraic group over k.

Definition 2.1.0.1. Let G be an algebraic scheme over k and let m : G×G→ G bea regular map. The pair (G,m) is an algebraic monoid over k if there exists a regularmap e : Spm(k)→ G such that the following diagrams commute

G×G×G id×m//

m×id

G×Gm

G×G m

// G,

Spm(k)×G e×id //

∼&&NN

NNNNNN

NNNN

G×Gm

G× Spm(k)id×eoo

∼xxppp

pppppp

ppp

G.

Furthermore, (G,m) is an algebraic group if it is an algebraic monoid and there existsa regular map inv : G→ G, such that the following additional diagram commutes

G(inv,id) //

G×Gm

G(id,inv)oo

Spm(k) e

// G Spm(k).eoo

On the other hand, (G,m) is a group variety when G is a variety, and when G is anaffine scheme, we will call (G,m) affine algebraic group. Also, ϕ : (G,m)→ (G′,m′) isa homomorphism of algebraic groups if ϕ : G→ G′ is a regular map and the followingdiagram commutes

G×G m //

ϕ×ϕ

G

ϕ

G′ ×G′m′

// G′.

In addition, we say that an algebraic group G is trivial if e : Spm(k) → G is anisomorphism, and a homomorphism ϕ : (G,m) → (G′,m′) is trivial if it factors

19

through e′ : Spm(k) → G′, that is, if there exists a morphism ψ : G → Spm(k) suchthat the following diagram commutes

Gϕ //

ψ ##GGG

GGGG

GGG′

Spm(k)

e′

OO

Definition 2.1.0.2. An algebraic subgroup of an algebraic group (G,mG) over k is analgebraic group (H,mH) over k such that H is a k-subscheme of G and the inclusionmap i : H → G is a homomorphism of algebraic groups, that is id m = m′ (i× i).A subgroup variety is an algebraic subgroup that is an algebraic variety.

The following result will provide us an intuition of algebraic groups.

Proposition 2.1.0.3. Let X be a algebraic k-scheme. There exists a bijective cor-respondence between the set of k-points X(k) and the set of scheme-theoretic pointsx ∈ |X| such that κ(x) = k.

Proof. Since any algebraic k-scheme can be covered by affines, it will suffice to showit for affine algebraic k-schemes. Thus, we can suppose X = Spm(A). We knowthat there exists a bijection between X(k) := HomAffSchk

(Spm(k), Spm(A)) andHomAlgk

(A, k), therefore, for each α ∈ X(k), there exists a morphism of k-algebrasα : A→ k. For m := ker(α), the diagram

Aα //

==

== k

Am

≀

OO

commutes. For the reverse inclusion, notice that if x is a scheme-theoretic pointwith residue field κ(x) = K, then it corresponds to a morphism Spm(K) → X. Inparticular, if κ(x) = k, there exists a morphism Spm(k)→ X.

Thanks to the previous proposition, it is clear that the map

m(k) : G(k)×G(k)→ G(k)

makes (G(k),m(k)) a group with neutral element e(k) and inverse map inv(k). Also,when k is algebraically closed, we know all of the maximal ideals of the k-algebra,therefore, G(k) = |G|, so (G,m) is a group where the maps x 7→ x−1 and x 7→ ax, fora ∈ G(k) are automorphisms of |G| as a topological space.This last point also shows that we can describe a homomorphism of algebraic groupsby describing its action on A-points. For instance, if we say that inv : G → G isx 7→ x−1, we mean that for all k-algebras A and all x ∈ G(A), inv(A)(x) = x−1.Another good example is the special linear group functor. If we define

SL(n, ·) := Spm(k[T11, . . . , Tnn]

⟨det(Tij)− 1⟩

),

20

together with the usual matrix multiplication

((aij), (bij)) 7→ (cij) =

(n∑l=1

ailblj

),

we have that SL(n, ·), that we will usually denote as SLn, is the algebraic group overk whose A-points are the matrices M ∈M(n,A) with det(M) = 1.Finally, an algebraic group (G,m) is said to be commutative whenever m t = m,where t is the transposition map G×G→ G×G such that (x, y) 7→ (y, x).

2.2 Basic properties of algebraic groupsNow let us see some of the functorial properties of algebraic groups. We know thatgiven an algebraic scheme X over k, we can define a functor X : Alg0

k → Set suchthat A 7→ X(A). We saw in the previous chapter that the functor

AlgSchk → Func(Alg0k,Set)

given by X 7→ X is fully faithful, thanks to Yoneda’s Lemma 1.2.0.4. We say that afunctor from k-algebras to sets is representable if it is of the form X. We want to showthat giving an algebraic group is equivalent to giving a functor in groups representedby it. In fact, this gives sense to the fact that in some books in the bibliography,affine algebraic groups are defined as functors from the category of k-algebras to thecategory of groups (for instance, see Chapter 6 of [Sza]).But that is easy to see, because one immediate consequence of the definitions of theresults from the previous section is that if we take (G,m) an algebraic group over k,then the functor

G : Algk → Grp

A 7−→ (G(A),m(A)),

where Grp denotes the category of groups, satisfies that every such functor arises froman essentially unique algebraic group, therefore, we can say that giving an algebraicgroup over k is equivalent to giving a functor Alg0

k → Grp such that its compositionwith the forgetful functor Grp→ Set is representable by an algebraic scheme. Thus,we can give an alternative definition of affine algebraic group.

Definition 2.2.0.1. An affine algebraic group G over k is a functor G : Algk → Grpsuch that its composition with the forgetful functor F : Grp→ Set is representableby some k-algebra A.

It is also important to observe that given G and H two algebraic k-groups, a morphismθ : G → H is in fact a natural transformation between the functors G,H : Algk →Grp. In particular, giving an affine algebraic monoid is the same as giving a functorAlgk →Mon, where Mon denotes the category of monoids, such that its compositionwith the forgetful functor is representable by an affine algebraic scheme.

21

Proposition 2.2.0.2. The maps e and inv from the definition of algebraic group areuniquely determined by (G,m). Also, if ϕ : (G,mG)→ (H,mH) is a homomorphismof algebraic groups, then ϕ eG = eH and ϕ invG = invH ϕ.

Proof. If we show the second statement, we will have the first one, because given analgebraic group (G,m) the fact that the map id : (G,m)→ (G,m) is a homomorphismof algebraic groups is trivial, thus if there were two identity elements e and e′, wewould have that e = id e = e′. Hence, let A be a k-algebra. Clearly,

(G(A),mG(A))→ (H(A),mH(A))

is a homomorphism of groups and so it maps the neutral element of G(A) to theneutral element of H(A). Also, it maps the inversion map of G(A) to the one onH(A). Eventually, Yoneda’s Lemma shows that the same is true for ϕ.

Proposition 2.2.0.3. Let G be an algebraic group and let H be a subscheme of G.H is an algebraic subgroup of G if and only if H(A) is a subgroup of G(A) for allk-algebras A.

Proof. Notice that if H(A) is a subgroup of G(A) for all A, then Yoneda’s Lemmashows that the maps

H(A)×H(A)→ H(A)

(h, h′) 7−−−−−−→ hh′

arise from a morphism mH : H×H → H such that (H,mH) is an algebraic subgroupof (G,mG). Let A be a k-algebra. The reverse implication can be seen by takingthe pair formed by H(A) and the morphism mH(A) : H(A)×H(A)→ H(A), whichforms a subgroup of G(A).

Proposition 2.2.0.4. Given any algebraic group (G,m), we have that the algebraicscheme G is separated.

Proof. All we have to do is remember theorem 1.3.2 from the previous section, that is,show that the diagonal ∆G(G) in G×G is closed, but this follows immediately fromthe fact that the preimage of e ∈ G(k) under the map m (id× inv) : (g, h) 7→ gh−1 is∆G(G), because (id× inv)−1 = id× inv therefore it is a closed set and we have it.

It is important to remark the fact that from now on, we will refer to an algebraicgroup G over k being irreducible, connected or geometrically conected indistinctly,because all three properties are equivalent.

Lemma 2.2.0.5. Let G be an algebraic group over k. The Zariski closure of asubgroup S of G(k) is a subgroup of G(k).

This can be shown using a few elementary results from a course in Topology. Fromthe lemma follows that

Theorem 2.2.0.6. Every algebraic subgroup of an algebraic group is closed.

22

Proof. Let H be an algebraic subgroup of an algebraic group G. If Hk is closed in Gk,then H is closed in G, therefore we can suppose that k = k without loss of generality.Also, we may suppose that H and G are reduced, because the transition from analgebraic group to its reduced algebraic subgroup does not change the underlyingtopological space.Thus, by definition |H| is locally closed or, equivalently, open in its closure, that wewill denote as S. Now, S is a subgroup of |G| by the previous lemma and it is a finitedisjoint union of cosets of |H|. Since each coset is open, it is also closed and thereforeH is closed in S from where we get the equality.

This shows that the algebraic subgroups of an algebraic group satisfy the topologicalnoetherian condition and also that every algebraic subgroup of an affine algebraicgroup is affine.Eventually, one can show that given an algebraic group G over k and S a closedsubgroup of G(k), there is a unique subgroup variety H of G such that S = H(k),and that the map from the set of subgroup varieties of G onto the set of closedsubgroups of G(k) given by H 7→ H(k) is a bijection.

2.3 Kernels and Group actionsDefinition 2.3.0.1. Let G be an algebraic group. We will say that

1. An algebraic subgroup H of G is normal if for any k-algebra A, H(A) ⊴ G(A).We will denote it as H ⊴ G.

2. An algebraic subgroup H of G is characteristic if for all k-algebras A and allα ∈ Aut(G(A)), α(H(A)) = H(A).

It is important to notice that thanks to Yoneda’s lemma, the condition that it musthold for all k-algebras can eventually be restricted to hold for all small k-algebras.Also, it can be shown that the identity component G of an algebraic group G ischaracteristic and therefore a normal subgroup of G.

Proposition 2.3.0.2. Let (G,m) be an algebraic group, let ϕ : G → H be a ho-momorphism of algebraic groups. We refer to the fiber product G ×H Spm(k) as thekernel of ϕ, which we denote as Ker(ϕ). It is an algebraic group, and it satisfies

Ker(ϕ)(A) = ker(ϕ(A)) ∀A ∈ Ob(Algk),

where ker has the usual definition of kernel in group theory.

Proof. Let A be a k-algebra. Notice that giving γ ∈ (G×H Spm(k))(A) is equivalentto giving morphisms α : Spm(A)→ G and β : Spm(A)→ Spm(k) so that the diagram

G ϕ

**TTTTTTT

TT

Spm(A)

α33hhhhhhhhh

β**UUUU

UUH,

Spm(k)e

55kkkkkk

23

commutes. Then α ∈ G(A) and ϕ(A)(α) = ϕα = e(Spm(A)), that is, α ∈ ker(ϕ(A)).Finally, a subscheme of an affine scheme is affine, therefore we have it.

Thanks to the first isomorphism theorem, we have that if G and H are affine, sinceN := Ker(ϕ) is also affine because of the proposition we showed that any subgroup ofan affine algebraic group is also affine and that the product of affine algebraic schemesis an affine algebraic scheme,

O(N) = O(G)⊗O(H) k ≃O(G)IHO(G)

,

where IH = ker(id : O(H) → k) and id is given by f 7→ f(e). IH is called theaugmentation ideal of H.From now on, we will simply call functor any functor from small k-algebras to sets,and group functor any functor from small k-algebras to groups.

Definition 2.3.0.3. An action of a group functor G on a functor X is a naturaltransformation µ : G × X → X such that for all k-algebras A, µA := µ(A) is anaction of G(A) on X(A), that is, if

1. µA(e, x) = x, for all x ∈ X(A) and

2. µA(gh, x) = µA(g, µA(h, x)), for all x ∈ X(A) and g, h ∈ G(A).

On the other hand,

Definition 2.3.0.4. An action of an algebraic group G on an algebraic scheme X isa regular map µ : G×X → X such that the following diagrams commute

G×G×X id×µ //

m×id

G×Xµ

G×X µ

// X

Spm(k)×X //

∼''OO

OOOOOO

OOOO

G×Xµ

X.

Also, for x ∈ X(k), the orbit map µx : G→ X such that g 7→ gx is µx := µ|G×x≃G.We will say that G acts transitively on X if G(k) acts transitively on X(k).

Clearly, because of Yoneda’s Lemma, giving the action of an algebraic group on analgebraic scheme is the same as giving an action of G on X.

2.4 Affine Algebraic groupsLet (G,m) be an affine algebraic k-group. Remember that, since G is affine, G =Spm(Γ(G,O)). By Proposition 1.1.2.9,

HomAlgSchk(G×G,G) ≃ HomAlgk

(Γ(G,O),Γ(G,O)⊗k Γ(G,O)),

24

therefore, the regular map m : G×G→ G induces a homomorphism of k-rings

∆ : O(G)→ O(G×G) ≃ O(G)⊗k O(G),

sending f 7→∑gi ⊗ hi, where f(xy) =

∑gi(x)hi(y). We call this morphism the

comultiplication map. On the other hand, the identity map e : Spm(k)→ G inducesa morphism ϵ : A → k, given by f 7→ f(e), which is called the coidentity map. Wewill study both maps in depth in the next chapter, when we introduce the notion ofHopf algebras.

Example 2.4.0.1. The most commonly known algebraic groups are the following.

1. The additive group is the functor

Ga : Algk → Grp

such that for any A ∈ Ob(Algk), Ga(A) := (A,+). Clearly, O(Ga) = k[X],and given x ∈ Ga(A), there exists a unique f ∈ HomAlgk

(k[X], A) such thatGa(k[T ]) → Ga(A) sends X to x. From this, we obtain the comultiplicationmap, ∆ : k[X]→ k[X]⊗ k[X], given by ∆(X) = X ⊗ 1 + 1⊗X.

2. On the other hand, the multiplicative group is the functor Gm that sends eachA to (A×, ·). Clearly, O(Gm) = k[X,Y ]/⟨XY − 1⟩ = k[X, 1

X] ⊆ k(X). Using

the results from the previous chapter, the comultiplication map ∆ : k[X, 1X]→

k[X, 1X]⊗ k[X, 1

X] is given by ∆(X) = X ⊗X.

3. Let n ∈ N. We can define the functor µn that sends A 7→ µn(A) := (x ∈A : xn = 1, ·). Clearly, O(µn) = k[X]/⟨Xn − 1⟩. It is easy to see that thecomultiplication map is induced by that of Gm.

4. Let m,n ∈ N and Mm×n be the functor given by A 7→ Mm×n(A), that is, theadditive group of m×n matrices with entries in R. Clearly, it can be representedby k[X11, . . . , Xmn]. It is important to notice that if we fix V a k-vector spaceand denote End(VA) as the set of A-linear endomorphisms for a k-algebra A,the functor EndV : A 7→ End(VA) satisfies that if dimV = n, then fixing acertain basis for V gives an isomorphism between the functors EndV ≃Mm×n.

In further chapters, we see more examples of algebraic groups.

2.4.1 The general linear group GLn

Apart from the previous examples, there is one that concerns us specially, becauseof its importance regarding representations of affine algebraic groups. That is, thegeneral linear group, which is a functor GLn : Algk → Grp such that

A 7→ GLn(A) := M ∈Mn×n(A) : det(M) ∈ A×.

It is clear that it can be represented by

O(GLn) =k[X11, . . . , Xnn, X]

⟨det((Xij))X − 1⟩.

25

Also, if we denote D(Xij) := det((Xij)), the comultiplication map is given by themap

∆ : k

[X11, . . . , Xnn,

1

D(Xij)

]→ k

[X11, . . . , Xnn,

1

D(Xij)

]⊗k[X11, . . . , Xnn,

1

D(Xij)

]such that for any 1 ≤ i, j ≤ n,

∆(Xij) =n∑k=1

Xik ⊗Xkj.

For the arbitrary dimension case, let V denote any k-vector space and define thefunctor GLV : A 7→ Aut(V ⊗k A), that we will study later. Clearly, if dim(V ) = n,both functors GLV and GLn are isomorphic.

Example 2.4.1.1. To end this section, here are some more examples of algebraick-subgroups of GLn.

1. The algebraic group of upper triangular matrices

Tn : A 7→ Tn(A) := M = (aij) ∈Mn×n(A) : aij = 0 ∀i > j.

2. The algebraic group of diagonal matrices

Dn : A 7→ Dn(A) := M = (aij) ∈Mn×n(A) : aij = 0 ∀i = j.

2.5 Homomorphisms and productsProposition 2.5.0.1. Let Gini=1 be a finite collection of k-algebraic groups. Then,G1 ×G2 × · · · ×Gn is an algebraic group. We will refer to it as the product of Gi.

Proof. All we have to do is notice that the functor

G1 × · · · ×Gn : Algk −−−−−−−−→ Set

A 7−−−→ G1(A)× · · · ×Gn(A)

is represented by G1 × · · · ×Gn.

If the Gi are all affine, we have previously seen that the product is also affine, hence

O(G1 × · · · ×Gn) ≃ O(G1)⊗ · · · ⊗ O(Gn).

In fact, if we let G1, G2 and H be algebraic groups together with homomorphismsϕi : Gi → H of algebraic groups, following the same reasoning that we followed inthe previous proposition, we can define the fibre product of G1 and G2 over H as thealgebraic group represented by the functor R 7→ G1(A)×H(A) G2(A). We will denoteit as G1 ×H G2. Also, in the affine case, that is, when G1, G2 and H are all affine,G1 ×H G2 is affine as well, hence

O(G1 ×H G2) ≃ O(G1)⊗O(H) O(G2).

26

2.5.1 The Frobenius mapOur goal is to extend the notion of Frobenius map to affine algebraic schemes. Inorder to do so, let k be a field such that Char(k) = p = 0. Also, let A be a k-algebraand

φA : A −−−→ A

a 7−−−→ ap.

Let (A, i : k → A) be the k-algebra A together with the k-algebra structure morphismand denote Aφ as the k-algebra (A,φ i). On the other hand, let G be an algebraick-group and let G(p) : Algk → Grp be the functor defined by A 7→ G(Aφ).

Proposition 2.5.1.1. Considering the previous notations and assumptions, if G isan affine algebraic k-group, then

Γ(G(p),O) = O(G)⊗k,φ k.

Proof. All we have to do is consider the following commutative diagrams

A

Γ(G,O)a

33gggggggggggggq// Γ(G,O)⊗k,φ k

b

77ppppppp

k

OO

φ // k

OO i

EE Spm(A)

b ''OOOOOO

a

++WWWWWWWWWWWWWWW

i

##

G×Spm(k) Spm(k) //

G

Spm(k) // Spm(k),

where a ∈ G(Aφ) and b ∈ HomAlgk(O(G)⊗k,φk,A), and notice that a, b and i are just

the associated morphisms of affine algebraic schemes, which are obtained by applyingthe functor Spm(−), and we know that the fibre product of affine algebraic schemesis an affine algebraic scheme, which finishes the proof.

In addition, if G is not affine, we can cover it with open affines, obtaining that it isalso an algebraic group. We can summarize everything in the following corollary.

Corollary 2.5.1.2. G(p) is an algebraic k-group.

Because of the equivalence between k-algebras and groups, φA induces a naturalhomomorphism of groups φ : A → Aφ, therefore it arises from a homomorphism ofalgebraic groups Φ : G → G(p). This morphism is called the Frobenius map. In fact,if we define Φn as the morphism Φn : G→ G(pn), then,

Proposition 2.5.1.3. The kernel of Φn : G → G(pn) is a characteristic subgroup ofG.

27

28

Chapter 3

Hopf Algebras

3.1 Bialgebras and affine monoidsThroughout this section, let k be a field. The notion of k-coalgebra that we are aboutto define can be seen intuitively as the dual notion of k-algebra. Firstly, though, letus give an alternative definition of k-algebra.

Definition 3.1.0.1. A k-algebra is a triple (A,∇, η), where A is a k-module, andthe maps ∇ : A ⊗k A → A and η : k → A, called multiplication and identity mapsrespectively, make the following diagrams commute

A⊗k A⊗k AidA⊗∇//

∇⊗idA

A⊗k A∇

A⊗k A ∇// A,

A⊗k kid⊗η //

∼&&LL

LLLLLL

LLLA⊗k A

∇

k ⊗k Aη⊗idoo

∼xxrrr

rrrrrr

rr

A

Also, if we let (A,∇, η) and (A′,∇′, η′) be k-algebras, a k-algebra homomorphismfrom A to B is a map ϕ : A→ B such that is k-linear and

1. ϕ ∇ = ∇′ (ϕ⊗ ϕ) and

2. ϕ η = η′.

It can be easily seen that the previous definitions coincide with the usual definitionof k-algebra and k-algebra homomorphism.

Definition 3.1.0.2. A k-coalgebra is a triple (B,∆, ϵ), where B is a k-module en-dowed with maps ∆ : B → B ⊗k B and ϵ : B → k, that we call comultiplication andcounit respectively, making the diagrams commute,

B∆ //

∆

B ⊗k B∆⊗idB

B ⊗k B idB⊗∆// B ⊗k B ⊗k B,

B

∆

∼

&&MMMMM

MMMMMM

∼

xxrrrrrr

rrrrr

B ⊗k k B ⊗k Bid⊗ϵoo

ϵ⊗id// k ⊗k B.

29

Furthermore, if we let τ ∈ Aut(B ⊗k B) such that τ : x⊗ y 7→ y ⊗ x, k-coalgebra Bis said to be co-commutative when τ ∆ = ∆. Let (B,∆B, ϵB) and (C,∆C , ϵC) bek-coalgebras. A map ϕ : B → C is a k-coalgebra homomorphism from B to C if it isk-linear and satisfies

1. (ϕ⊗ ϕ) ∆B = ∆C ϕ and

2. ϵB = ϵC ϕ.

Remark 3.1.0.3. It is worth mentioning how to build a coalgebra given a pair ofcoalgebras. Assume (B,∆, ϵ) and (B′,∆′, ϵ′) are k-coalgebras. Since k ≃ k ⊗k k, wecan construct a counit by defining the map

k ≃ k ⊗k kϵ⊗ϵ′−−→ B ⊗k B′.

For the comultiplication, if we let τ ∈ Hom(B⊗kB′, B′⊗kB) that sends b⊗b′ 7→ b′⊗b,then we have

B ⊗k B∆⊗∆′

//

∆ ))SSSSSSSS B ⊗k B ⊗k B′ ⊗k B′

idB⊗τ⊗idB′

(B ⊗k B′)⊗k (B ⊗k B′),

where ∆ = (idB ⊗ τ ⊗ idB′) (∆⊗∆′). Thus, since B ⊗k B′ is a k-module, we havethat the triple (B⊗kB′,∆, ϵ⊗ ϵ′) is a k-coalgebra. We can follow a similar argumentin order to obtain a new k-algebra from a pair of k-algebras.

Example 3.1.0.4. Some of the most common examples of coalgebras are the follow-ing.

1. The polynomial coalgebra over a field k consists on the k-module k[X] and themaps

∆ : k[X] −→ k[X]⊗k2

Xn 7−→ Xn ⊗Xn

ϵ : k[X] −→ k

Xn 7−→ 1.

It is easy to verify that ∆ and ϵ make the diagrams from 3.1.0.2 commute. Itis also cocommutative.

2. Assume that eij1≤i,j≤n is the canonical k-basis for Mn(k). The matrix coal-gebra is the coalgebra given by

∆ :Mn(k)→ Mn(k)⊗k2

eij 7→∑

eik ⊗ ekjϵ :Mn×n(k) −→ k

eij 7−→ δij,

where δij denotes the Kronecker delta function. Clearly, it is a k-coalgebra.

It is easy to verify that the previous definitions translate into the following lemma.

Lemma 3.1.0.5. Let B be a k-module endowed with a multiplication ∇ : B⊗kB →B, a unit η : k → B, a comultiplication ∆ : B → B ⊗k B and a counit ϵ : B → k.Suppose that the triple (B,∇, η) is a k-algebra and that (B,∆, ϵ) is a k-coalgebra.Then, it is equivalent to say that

30

1. ∆ and ϵ are morphisms of k-algebras,

2. ∇ and η are morphisms of k-coalgebras.Finally, we can give the definition of a k-bialgebra.Definition 3.1.0.6. A k-bialgebra is a 5-tuple (A,∇, η,∆, ϵ) such that (A,∇, η) is ak-algebra, (A,∆, ϵ) is a k-coalgebra, and one of the two equivalent conditions of theprevious lemma is satisfied.

3.2 Hopf AlgebrasDefinition 3.2.0.1. Let (B,∇,∆, η, ϵ) be a k-bialgebra. An antipode is a morphismof k-algebras S : B → B such that the following diagram commutes

B ⊗k Bid⊗S // B ⊗k B ∇

((QQQQQ

QQ

B

∆ 66mmmmmmm

∆((QQ

QQQQQ

ϵ // kη // B

B ⊗k B S⊗id// B ⊗k B

∇

66mmmmmmm

A Hopf algebra is a k-bialgebra that admits an antipode. Let A and B be Hopfk-algebras. A homorphism of Hopf k-algebras f : A → B is a homomorphism ofk-algebras such that

(f ⊗ f) ∆A = ∆B f.Also, we say that a k-Hopf algebra is commutative if it is a commutative k-algebra,and cocommutative if it is a cocommutative k-coalgebra.

Now, let A be a k-algebra and ∆ : A→ A⊗A a homomorphism of k-algebras. Takinginto account the properties of Spm that we saw in the first chapter, we have that

Spm(∆) : Spm(A⊗ A) ≃ Spm(A)× Spm(A)→ Spm(A).

The following proposition shows the duality between affine algebraic groups and Hopfalgebras.Proposition 3.2.0.2. Given a k-algebra (A,∇, η), a 6-tuple (A,∇, η,∆, ϵ, S) is aHopf k-algebra if, and only if, the 4-tuple (Spm(A), Spm(∆), Spm(ϵ), Spm(S)) is analgebraic group.Proof. Set G := Spm(A), m := Spm(∆), e := Spm(ϵ) and inv := Spm(S). Then,the commutativity of the left diagram implies the comutativity of the right one, andviceversa

G×G×G idG×e //

e×idG

G×Gm

G×G m

// G,

A⊗k A⊗K A⊗k AidA⊗ϵoo

A⊗k A

∆

OO

A,

ϵ⊗idA

OO

∆oo

because of the duality between Algk and AlgSchk. The rest of the proof proceedsthe same way, by comparing the corresponding commutative diagrams.

31

Corollary 3.2.0.3. The functor Spm is an equivalence of categories between thecategory Hopfk of Hopf k-algebras and AffAlgGrpk, with quasi inverse (G,m) 7→(Γ(G,O),O(m)).

The previous corollary gives us a lot of examples of Hopf algebras, all we have todo is take an affine algebraic group and give the morphisms that make its associatedk-module A a Hopf k-algebra. Example 2.4.0.1 gives us the most important examplesand how to find the Hopf algebra associated to an affine algebraic group. Either way,let us show another interesting example of Hopf algebras.

Example 3.2.0.4. Remember that a Lie algebra g is a vector space over a field kwith an operation [−,−] : g×g→ g which is bilinear, antisymmetric and that satisfiesthe Jacobi identity, that is, ∀x, y, z ∈ g,

[x, [y, z]] = [[x, y], z] + [y, [x, z]].

Also, remember that since g is a k-vector space, we can construct the tensor algebraassociated to g as

T (g) := k ⊕⊕n≥1

g⊗kn

and then, the universal algebra associated to g, U(g), is given by taking the quotient

U(g) :=T (g)

I,

where I = ⟨a⊗ b− b⊗ a− [a, b] : a, b ∈ g⟩. It turns out that the universal envelopingalgebra U(g) is a Hopf algebra when it is endowed with the obvious structure of k-algebra and the k-coalgebra structure given by the coproduct ∆(x) = x⊗ 1 + 1⊗ x,the counit ϵ(x) = 0 and antipode S(x) = −x where x ∈ g extended to U(g) in thenatural way. It is also clear that it is commutative if and only if g is abelian and thatit is cocommutative.

3.2.1 Hopf subalgebrasDefinition 3.2.1.1. A k-subalgebra B of a Hopf Algebra (A,∆, ϵ, S) is a Hopf sub-algebra if ∆(B) ⊆ B ⊗k B and S(B) ⊆ B.

The following proposition shows some properties of Hopf k-subalgebras.

Proposition 3.2.1.2. Let (A,∆A, ϵA, SA) and (B,∆B, ϵB, SB) be two Hopf k-algebrasand f : A→ B a homomorphism of Hopf algebras. Then,

1. If C is a Hopf k-subalgebra of A, then (C,∆A|C) is a Hopf k-algebra.

2. The image f(A) is a Hopf k-subalgebra of B.

Definition 3.2.1.3. A Hopf ideal in a Hopf k-algebra (A,∆, ϵ,∇, η, S) is an ideal ain A such that ϵ(a) = 0, S(a) ⊆ a and

∆(a) ⊆ A⊗ a+ a⊗ A.

32

In fact, the following proposition shows that some results from linear algebra can beextended to Hopf k-algebras.

Proposition 3.2.1.4. Let A and B be Hopf k-algebras and let f : A → B be amorphism of Hopf k-algebras. The kernel and the image of a homomorphism of Hopfk-algebras are Hopf ideals. Also,

1. If a ⊆ A is a Hopf ideal, the quotient vector space A/a has a unique Hopfk-algebra structure for which the canonical morphism ϕ : A→ A/a is a homo-morphism of Hopf k-algebras. In addition, if a ⊆ ker(f), f factors uniquelythrough ϕ.

2. The homomorphism f induces an isomorphism of Hopf k-algebras,

f :A

ker(f)−−→ im(f).

And finally, if we let G be an affine algebraic k-group, we can give the relationshipbetween Hopf k-subalgebras of O(G) and of algebraic subgroups of G.

Proposition 3.2.1.5. Let G be an affine algebraic k-group. In the bijective corre-spondence between closed subschemes of G and ideals in O(G), algebraic subgroupscorrespond to Hopf ideals.

33

34

Chapter 4

Linear representations andcharacters of algebraic groups

4.1 RepresentationsHenceforth, G will denote an affine algebraic k-group and V a k-module. Also,remember that GLV denotes the functor

GLV : Algk −−→ Grp

A 7−−−→ Aut(V ⊗k A).

If V is a finite dimensional k-vector space, then GLV is given by

A 7→ GL(dim(V ), A),

and also let us define functor WV : Algk →ModA by WV (A) := V ⊗k A.

Definition 4.1.0.1. A linear representation of G is a pair (V,Φ), where V is a k-module and Φ is a natural transformation, Φ : G→ GLV , such that the component ofΦ at each k-algebra A is a homomorphism of groups. We say that Φ is faithful if Φ(A)is injective for any A ∈ Ob(Algk). Sometimes we will denote linear representationswith caligraphic letters, for instance, M will denote (M,Φ). In addition, if M andN are representations of G, a G-homomorphism f : M → N consists of a naturaltransformation f : WM →WN such that the diagram

WM(A)ΦM(A)(g)//

f(A)

WM(A)

f(A)

WN(A)ΦN (A)(g)

// WN(A)

commutes for every A ∈ Ob(Algk) and g ∈ G(A). We denote the category of repre-sentations of an algebraic group G as Rep(G) and the set of G-homomorphisms isdenoted as HomG(M,N ) or HomRep(G)(M,N ).

35

The following proposition gives us the relationship between actions and representa-tions.

Proposition 4.1.0.2. Given a k-module V , there is a bijective correspondence be-tween the set of actions of an affine algebraic k-group G on WV and the set of linearrepresentations of G on V .

Proof. Suppose that Φ : G → GLV is a linear representation and let A be a k-algebra. Then, we can define an action µ : G ×WV → WV by setting, for eachx = v ⊗ a ∈ WV (A) = V ⊗k A and each g, h ∈ G(A), µ(A)(g, x) := Φ(A)(g)(v ⊗ a).Clearly, µ(A)(e, v⊗a) = v⊗a and µ(A)(gh, v⊗a) = µ(A)(g, µ(A)(h, v⊗a)), hence itis indeed an action. For the reverse implication, if µ is an action of the group functorG on WV , then we can define, for each A ∈ Ob(Algk),

Φ(A)(g)(v ⊗ a) := µ(A)(g, v ⊗ a), ∀v ⊗ a ∈ V ⊗k A, g ∈ G(A).

The result follows from the properties of µ(A), that is, if we translate the diagramsthat commute from the definition of µ, we get the result we want.

This shows that giving a representation (V,Φ) is equivalent to giving an action of Gon WV . When we see the representation as an action, we will refer to (V,Φ) as aG-module.

Example 4.1.0.3. Let us show some important examples of representations.

1. Let V = k and let A ∈ Ob(Algk). Take the morphism

Φ(A) : G(A) −−→ GLk(A) = A×

g 7−−−→ 1A.

Clearly, this is a representation, and we refer to it as the trivial representationof G. In many books, it is written as triv.

2. Let M,N ∈ Ob(Rep(G)). It is easy to see that for any k-algebra A, thereis a natural isomorphism WM(A) ⊗A WN(A)

∼−→ WM⊗kN(A). Therefore, wecan define an action of G(A) on WM(A)⊗AWN(A) by setting ΦM⊗kN(A)(g) =ΦM(A)(g) ⊗A ΦN(A)(g). It is also easy to check that this is a representationof G, and we denote it as M⊗G N ∈ Ob(Rep(G)). It will be of great use forthe next chapter to notice that the natural k-isomorphism M ⊗kN → N ⊗kMgives an isomorphism

βM,N :M⊗G N∼−→ N ⊗GM∈ HomRep(G)(M⊗G N ,N ⊗GM),

such that βN ,M βM,N = idM⊗GN , and that for a pair of morphisms F ∈HomRep(G)(M,M′) and F ′ ∈ HomRep(G)(N ,N ′), the morphism F ⊗G F ′ ∈HomRep(G)(M⊗G N ,M′ ⊗G N ′) is defined naturally. In the next chapter, wesee that this translates into the fact that Rep(G) is a symmetric monoidalcategory.

36

3. LetM∈ Ob(Rep(G)). Take the dual of M , M∨ := HomModk(M,k), and note

that for every k-algebra A, there is an isomorphism HomModA(WM(A), A) →

WM(A)∨, thus we can define an action of G on M∨ by taking g ∈ G(A),x ∈WM(A) and setting (gϕ)(x) := ϕ(g−1(x)). This is a representation of G onM∨, denoted M∨ ∈ Ob(Rep(G)).

4. Finally, for M,N ∈ Ob(Rep(G)), using the previous constructions and thenatural isomorphism M∨ ⊗k N → HomModk

(M,N) (see [Lan]), we have a rep-resentation denoted as Homk(M,N ).

Definition 4.1.0.4. Let G be an affine algebraic k-group. A Γ(G,O)-comodule is ak-linear map ρ : V → V ⊗k Γ(G,O) such that the following diagrams commute

Vρ //

ρ

V ⊗k Γ(G,O)idV ⊗∆

V ⊗k Γ(G,O)ρ⊗idΓ(G,O)

// V ⊗k Γ(G,O)⊗ Γ(G,O),

Vρ //

LLLLLL

LLLLLL

LLLLLL

LLLLLL

V ⊗k Γ(G,O)idV ⊗ϵ

V ≃ V ⊗k k.

Furthermore, a Γ(G,O)-subcomodule of the Γ(G,O)-comodule (V, ρ) is a k-subspaceW ⊆ V such that

ρ(W ) ⊆ W ⊗k Γ(G,O).

Notice that if W is a Γ(G,O)-subcomodule, (W, ρ|W ) is in fact a Γ(G,O)-comodule,because the fact that ρ makes the previous diagrams commute implies that the re-striction of ρ to W makes the diagrams commute as well.

Now let us see that the set of linear representations of an affine algebraic k-group Gon V is not only in bijection with the set of actions of G on WV , but also with theset of Γ(G,O)-comodule structures on V .

Theorem 4.1.0.5. Let G be an affine group scheme. The set of linear representationsof G on V is in a bijective correspondence with the set of Γ(G,O)-comodule structureson V .

Proof. Let Φ : G → GLV be a representation of G, and suppose that G = Spm(A).For the general element id ∈ G(A), applying the natural transformation Φ, we get anA-linear map Φ(id) : Φ(A) = V ⊗kA→ Φ(A) = V ⊗kA such that it is determined byits restriction to V , that is, we can define the restriction ρ := Φ(id)|V⊗kk : V ⊗k k ≃V → V ⊗k A = V ⊗k Γ(G,O). On the other hand, thanks to Yoneda’s lemma andthe fact that Φ is a natural transformation, for each R ∈ Ob(Algk) and for eachg ∈ G(R) ≃ HomAlgk

(A,R), the following diagram commutes

V ⊗k AΦ(id) //

idV ⊗g

V ⊗k AidV ⊗g

V ⊗k RΦ(g)

// V ⊗k R.

37

Therefore, since Φ(g) acts by (id⊗ g) ρ, we have shown that Φ is determined by ρ.For the reverse implication, we know that for any k-linear map ρ : V → V ⊗k A wecan get a natural map between sets, namely

Φ : G(R)→ EndV (V ⊗k R).

We must show that the unit in G(R) acts as the identity, that is, for all R ∈ Ob(Algk),the diagram

Vρ //

≃))SSS

SSSSSSSS

SSSSSSSS V ⊗k A = V ⊗k Γ(G,O)

id⊗ϵ

V ⊗k R V ⊗ koo

must commute for all R ∈ Ob(Algk), which is equivalent to the second diagram indefinition 4.1.4. Finally, we must show that Φ(g)Φ(h) = Φ(gh), but since gh is givenby

A∆−→ A⊗k A

(g,h)−−→ R,

on V the action Φ(g)Φ(h) is given by

Vρ−→ V ⊗k A

id⊗h−−→ V ⊗k Rρ⊗id−−→ V ⊗k A⊗k R

id⊗(g,id)−−−−−→ V ⊗k R

which coincides with

Vρ−→ V ⊗k A

id⊗∆−−−→ V ⊗k A⊗k Aid⊗(g,h)−−−−→ V ⊗k R

for all g and h if and only the first diagram of definition 4.1.0.4 commutes. Thus, wehave that the two diagrams from 4.1.0.4 commute and we have shown the result.

4.2 Properties of Γ(G,O)-comodulesFirstly, let (vi)i∈I be a basis of V and let (rij)i,j∈I be a family of elements of Γ(G,O).Our first goal is to translate the conditions of a k-linear map ρ : V → V ⊗k Γ(G,O)into a more explicit expression. Notice that if we want the map

ρ : V −−→ V ⊗k Γ(G,O)

vj 7−→∑i∈I

vi ⊗ rij

to be a Γ(G,O)-comodule, it must make the diagrams from definition 4.1.0.4 com-mute. More specifically, we can translate the two commutative diagrams into thefollowing relations

(idV ⊗∆) ρ = (ρ⊗ idΓ(G,O)) ρ (idV ⊗ ϵ) ρ = idV→V⊗kk.

Therefore, for i ∈ I, the following expressions must be equal

(idV ⊗∆) ρ(vj) = (idV ⊗∆)

(∑i∈I

vi ⊗ rij

)=∑i∈I

vi ⊗∆(rij)

38

(ρ⊗ idΓ(G,O))

(∑i∈I

vi ⊗ rij

)=∑i∈I

ρ(vi)⊗ rij =∑i,k∈I

vk ⊗ rki ⊗ rij,

and also the following equality must hold

(idV ⊗ ϵ)

(∑i∈I

vi ⊗ rij

)=∑i∈I

vi ⊗ ϵ(rij) = vj ⊗ 1 = idV→V⊗kk(vj),

which is equivalent to

1. ∆(rij) =∑

k∈I rik ⊗ rkj and

2. ϵ(rij) = δij.

This construction is of great use, because it makes some of the results ahead more easyto be worked with. For instance, in the next section we see an interesting propertyof comodules, which is proved using the previous notation.Now, our next goal is to show that every representation can be seen as a unionof finite-dimensional representations. In order to do so, we are going to need thefollowing proposition, which will characterize what we call group stabilizers. Theproof is left to the reader, as it is a mere checking.

Proposition 4.2.0.1. Let Φ : G→ GLV be a finite-dimensional representation of Gand let W be a subspace of V . The functor given by

StabG(W ) : Algk −−−−→ Grp

A 7−→ StabG(W )(A),

where StabG(W )(A) := g ∈ G(A) : g(WW (A)) = WW (A), is represented by analgebraic subgroup StabG(W ) of G.

We call the subgroup StabG(W ) the stabilizer of W in V , and say that an algebraicsubgroup H of G stabilizes a subspace W of V if it satisfies H ⊂ StabG(W ).

Proposition 4.2.0.2. A subspace W of a k-vector space V is stable under G if andonly if it is a Γ(G,O)-subcomodule of V

Proof. Thanks to Theorem 4.1.0.5, we know that there exists a Γ(G,O)-comoduleassociated to the representation. All we have to do is notice that the conditionG ⊂ StabG(W ) is equivalent to the condition that ρ(W ) ⊆ W ⊗k Γ(G,O).

The following proposition will be the last step towards our initial goal.

Proposition 4.2.0.3. Every Γ(G,O)-comodule (V, ρ) is a filtered union of its finite-dimensional sub-comodules.

39

Proof. It is known that a finite sum of finite-dimensional sub-comodules is a finite-dimensional sub-comodule, hence we only need to show that each element v ∈ V iscontained in a finite-dimensional sub-comodule. Therefore, let (ei)i∈I be a basis forΓ(G,O) as a k-module and

ρ(v) :=∑i∈I

vi ⊗ ei,

where vi ∈ V and all but finitely many vi are zero. Also, let

∆(ei) :=∑j∈I

∑l∈I

rijl(ej ⊗ el),

where rijl ∈ k. our goal is to show that

ρ(vl) =∑i∈I

∑j∈I

rijl(vi ⊗ ej),

because from that it follows that the k-submodule W of V spanned by v and the vi is asubcomodule containing v, that is, we would be showing that ρ(W ) ⊂ W ⊗k Γ(G,O).In order to do so, all we have to do is translate the commutativity of the first diagramin definition 4.1.4 as (idv ⊗∆) ρ = (ρ⊗ idΓ(G,O)) ρ. If we apply this last equalityto v, we get that∑

i,j,l∈I

rijl(vi ⊗ ej ⊗ el) =∑l∈I

ρ(vl)⊗ el ∈ V ⊗k Γ(G,O)⊗k Γ(G,O),

which implies the equality that we are seeking by comparing the coefficients in eachof the two expressions.

Corollary 4.2.0.4. Every representation of G is a filtered union of its finite-dimensionalsubrepresentations

Proof. Let Φ : G → GLV be a representation of G and ρ : V → V ⊗k Γ(G,O) itscorresponding co-action. All we have to do is apply the previous proposition and weget the result we want.

4.3 CharactersThe reader may have noticed at this point that most of the notions that we havefrom group theory can be extended to algebraic groups. Of course, character theoryis not an exception. In this section we show an important result regarding algebraiccharacters, the character decomposition theorem.

Definition 4.3.0.1. Given an affine algebraic group G, a character of G is a homo-morphism χ : G→ Gm.

The following proposition summarizes some of the most common ways of workingwith characters.

40

Proposition 4.3.0.2. Let G be an affine algebraic k-group. Giving a character χ ofG is equivalent giving an element a = a(χ) ∈ Γ(G,O) such that ∆(a) = a ⊗ a. Anyelement satisfying that property is called group-like.

Proof. The result follows directly from the fact that Γ(Gm,O) and ∆(X) = X ⊗X.

Definition 4.3.0.3. Let χ be a character of an affine algebraic k-group G. We saythat G acts on V through χ if there exists a representationM = (M,ΦM) such that

ΦM(R)(g)(v ⊗ r) = χ(g)(v ⊗ r), ∀g ∈ G(R), v ⊗ r ∈WV (R).

4.3.1 Eigenspaces and the decomposition theorem

The following proposition will introduce us to how we can study eigenspaces of groupswith certain characters.

Proposition 4.3.1.1. Let (V,Φ) be a representation of G and let ρ := ρΦ its corre-sponding Γ(G,O)-comodule. For any character χ : G→ Gm of G, there is a greatestsubspace Vχ of V on which G acts through χ. Furthermore,

Vχ = v ∈ V : ρ(v) = v ⊗ a(χ).

Proof. For the first part of the proof, we say that G acts on a subspace W of Vthrough a character χ if W is stable under G and G acts on W through χ. Thistranslates into the fact that if w ∈ W , then for every g ∈ G(k), Φ(g)w = λw for acertain λ ∈ k, that is, the elements of W are common eigenvectors for the g ∈ G(k).If G acts on subspaces W and W ′ through a character χ, the it acts on W + W ′

through χ because of the linearity. This shows the first point of the proposition, i.e,there is a greatest subspace Vχ of V on which G acts through χ. For the second pointof the proof, the set

v ∈ V : ρ(v) = v ⊗ a(χ)is a subspace of V on which G acts through χ, and the fact that it contains everysuch subspace implies the result we want to show.

The subspace Vχ is called the eigenspace for G with character χ. In addition to this,the following lemma will be of great use in order to show the decomposition theorem.

Lemma 4.3.1.2. The group-like elements in a Hopf k-algebra (A,∆, ϵ,∇, η, S) arelinearly independent.

Proof. Firstly, it is important to notice that for a given Hopf algebra A together witha group-like element a ∈ A, translating the commutative diagrams of 3.1.0.2

1⊗ a = idA→A⊗kk(a) = ((ϵ⊗ idA) ∆)(a) = (ϵ⊗ idA)(a⊗ a) = ϵ(a)⊗ a,

which implies that ϵ(a) = 1. Secondly, suppose that the group-like elements in A arenot linearly independent. Then, it is possible to express one group-like element e as

e =∑i∈I

ciei,

41

where ci ∈ k and ei = e. Actually, we may even suppose that the ei are linearlyindependent, which gives us that the ei ⊗ ej are also linearly independent. We have

∆(e) = e⊗ e =∑i,j∈I

cicjei ⊗ ej ∆(e) =∑i∈I

ci∆(ei) =∑i∈I

ciei ⊗ ei,

where the left equality is clear and the right one follows from the fact that ∆ is ak-algebra morphism and the ei are group-like. Thanks to the previous equalities,

cicj = ciδij

for all i, j ∈ I. Now, thanks to the first observation, since ϵ(e) = 1,

1 = ϵ(e) =∑i∈I

ciϵ(ei) =∑i∈I

ci,

so the ci form a complete set of orthogonal idempotents in the field k. Hence, oneof them must be equal to one and the rest equal to zero, which contradicts ourassumption that e is not equal to any of the ei.

All we have to do now is prove the theorem, which follows directly from the lemma.

Theorem 4.3.1.3 (Decomposition in characters). Let Φ : G → GLV be a represen-tation of an affine algebraic group on a vector space V . If V is a sum of eigenspaces,for instance V =

∑χ∈Ξ Vχ, where Ξ is a set of characters of G, then

V =⊕χ∈Ξ

Vχ

Proof. If the sum is not direct, there exists a finite set of characters χ1, . . . , χm anda relation

m∑i=1

vi = 0,

where vi ∈ Vχiand vi = 0. When we apply ρ to this relation, the fact that Vχi

=v ∈ V : ρ(v) = v ⊗ a(χi) implies that

0 = ρ

(m∑i=1

vi

)=

m∑i=1

ρ(vi) =m∑i=1

vi ⊗ a(χi).

Also, for every linear map f : V → k,∑m

i=1 f(vi)a(χi) = 0, which contradicts thelinear independence of the a(χi) and therefore Lemma 4.3.1.2.

42

Chapter 5

Tannakian Categories

In this section, our main goal is to introduce the notion of a Tannakian category. Itwill also serve as a prelude in order to introduce several notions that come in handythroughout this last part of the work.

5.1 Abelian CategoriesDefinition 5.1.0.1. Let C be a category and let X,Y ∈ Ob(C). We define a productof X and Y as an object X × Y ∈ Ob(C) together with morphisms p ∈ HomC(X ×Y,X) and q ∈ HomC(X × Y, Y ) such that for all W ∈ Ob(C) and morphisms α ∈HomC(W,X) and β ∈ HomC(W,Y ) there is a unique γ ∈ HomC(W,X × Y ) makingthe next diagram commute.

Wβ

((γ ((P

PPPPPP

α

%%

X × Yp

q// Y

X

We say that C has products of pairs of objects if a product X × Y exists for anyX,Y ∈ Ob(C).Definition 5.1.0.2. Let C be a category and let X,Y be a pair of objects in C.A coproduct of X and Y is an object X ⨿ Y ∈ Ob(C) together with morphismsi ∈ HomC(X,X ⨿ Y ) and j ∈ HomC(Y,X ⨿ Y ) such that for any W ∈ Ob(C) andmorphisms α ∈ HomC(X,W ) and β ∈ HomC(Y,W ) there is a unique γ ∈ HomC(X ⨿Y,W ) making the diagram

Y

j β

X i //

α--

X ⨿ Yγ

((PPPPPPP

W

43

commute. Also, we say that the category C has coproducts of pairs of objects if acoproduct X ⨿ Y exists for any X,Y ∈ Ob(C).

Definition 5.1.0.3. Let C be a category. We say that X ∈ Ob(C) is an initial objectif for every object Y of C there is exactly one morphism X → Y . On the other hand,X is called a final object if for every object Y ∈ Ob(C) there is exactly one morphismY → X.

Definition 5.1.0.4. A category A is called preadditive if each morphism set HomA(X,T )is endowed with the structure of an abelian group such that the compositions

HomA(X,Y )× HomA(Y, Z) −→ HomA(X,Z)

are bilinear. A functor F : A → B of preadditive categories is called additive if andonly if F : HomA(X,Y )→ HomB(F (X), F (Y )) is a homomorphism of abelian groupsfor all X,Y ∈ Ob(A).

Lemma 5.1.0.5. Let A be a preadditive category and let X be an object of A. Thefollowing are equivalent

1. X is an initial object,

2. X is a final object, and

3. idX = 0 in HomA(X,X).

Any object that is both final and initial is called zero object, and is denoted by 0.

The proof of 5.1.0.5 is left as an exercise for the reader, all that needs to be doneis apply the fact that each HomA(X,Y ) has the structure of an abelian group andthe result easily follows.

Lemma 5.1.0.6. Let A be a preadditive category and let X,Y ∈ Ob(A). Then,X × Y exists if and only if X ⨿ Y does. In addition, if any of them exists, thenX × Y ≃ X ⨿ Y .

Proof. Suppose that X × Y exists, together with projections p : X × Y → X andq : X × Y → Y . Let i : X → X × Y be the morphism corresponding to (1, 0) andj : Y → X×Y the morphism corresponding to (0, 1). Thus we have the commutativediagram

X1 //

i

##HHH

HHHH

HHX

i

##HHH

HHHH

HH

X × Y

p;;vvvvvvvvv

q

##HHH

HHHH

HHip+jq //_______ X × Y

Y1 //

j;;vvvvvvvvv

Yj

;;vvvvvvvvv

where the diagonal compositions are zero. It follows that ip+j q : X×Y → X×Yis the identity since it is a morphism such that when it is composed with p it givesp and when it is composed with q gives q. Let a : X → W and b : Y → W be two

44

morphisms. Then we can form the map ap+ b q : X×Y → W . In this way we geta bijection HomA(X × Y,W ) ≃ HomA(X,W ) × HomA(Y,W ) from which we obtainthat that X × Y ≃ X ⨿ Y . The other case, that is, the one in which we work with acoproduct, can be easily seen using an analogous reasoning.

Definition 5.1.0.7. Let A be a preadditive category and let X,Y ∈ Ob(A). Theproduct X × Y endowed with the morphisms i, j, p, q from the proof of the previouslemma is called direct sum and is denoted as X⊕Y . Furthermore, A is called additiveif it is preadditive and finite products exist, that is, it has a zero object and directsums.

Definition 5.1.0.8. Let A be a preadditive category. Let f ∈ HomA(X,Y ).

1. A kernel of f is a morphism i : Z → X such that

(a) f i = 0 and(b) for any i′ : Z ′ → X such that f i′ = 0 there exists a unique morphism

g : Z ′ → Z such that i′ = i g.

If a kernel of f exists, then it is unique, up to a unique isomorphism and wedenote it Ker(f)→ X.

2. We define a cokernel of f as a morphism p : Y → Z such that

(a) p f = 0 and(b) for any p′ : Y → Z ′ such that p′ f = 0 there exists a unique morphism

g : Z → Z ′ such that p′ = g p.

If a cokernel of f exists, then it is unique, up to a unique isomorphism and wedenote it Y → Coker(f).

3. If a kernel of f exists, then a coimage of f is a cokernel for the morphismKer(f)→ X. If a kernel and a coimage exist then we denote this X → Coim(f).

4. If a cokernel of f exists, then the image of f is a kernel of the morphismY → Coker(f). If a cokernel and an image of f exist then we denote thisIm(f)→ Y .

Lemma 5.1.0.9. Let A be a preadditive category, let X,Y ∈ Ob(A) and let f ∈HomA(X,Y ) such that kernel, cokernel, image and coimage exist. Then, f can befactored uniquely as

Xf //

Y

Coim(f) // Im(f).

OO

45

Proof. Due to the definition of kernel, we know ker(f) → X → Y is zero, thuswe have a canonical morphism Coim(f) → Y . On the other hand, Coim(f) →Y → Coker(f) is zero, because it is the unique morphism that gives rise to X →Y → Coker(f), which is zero. Therefore, Coim(f) → Y factors uniquely throughIm(f)→ Y , therefore we have the map we are looking for.

Definition 5.1.0.10. A category A is abelian if it is additive, if all kernels and coker-nels exist, and if the natural map from 5.1.0.9, Coim(f)→ Im(f), is an isomorphismfor all morphisms f of A.

5.2 Symmetric Monoidal CategoriesDefinition 5.2.0.1. Let C be a category. C is a k-linear category if every morphismset is a k-module and for every X,Y, Z ∈ Ob(C), the composition law

HomC(Y, Z)× HomC(X,Y )→ HomC(X,Z)

is k-bilinear.

Definition 5.2.0.2. A monoidal category C = (C,⊗, α,1, l, r) consists on the follow-ing data

1. A k-linear category C,

2. A functor ⊗ : C× C→ C called the tensor product of C,

3. An isomorphism of functors α : ⊗ (idC ×⊗)→ ⊗ (⊗× idC) and

4. An object 1 ∈ Ob(C) and isomorphisms l : idC → 1⊗ idC and r : idC → idC⊗ 1,called right unit and left unit, respectively,

And such that the previous data satisfies the pentagon axiom, that is, for allX,Y, Z,W ∈Ob(C), the following diagram commutes

W ⊗ (X ⊗ (Y ⊗ Z))idW⊗αX,Y,Z

ttiiiiiiii

iiiiiiii

iαW,X,Y ⊗Z

**UUUUUUU

UUUUUUUU

UU

W ⊗ ((X ⊗ Y )⊗ Z)αW,X⊗Y,Z

(W ⊗X)⊗ (Y ⊗ Z)αW⊗X,Y,Z

(W ⊗ (X ⊗ Y ))⊗ Z

αW,X,Y ⊗idZ // ((W ⊗X)⊗ Y )⊗ Z.

Furthermore, if we let C = (C,⊗C, αC,1C, l

C, rC) and D = (D,⊗D, αD,1D, l

D, rD) bemonoidal categories, a monoidal functor C→ D is a triple (F, J, J0) such that

1. F : C→ D is a functor,

2. J : ⊗D (F × F )→ F ⊗C is a functor isomorphism and

46

3. J0 : F (1C)→ 1D is an isomorphism

and for every X,Y, Z ∈ Ob(C), the following diagrams commute

F (X)⊗D (F (Y )⊗D F (Z))idF (X)⊗DJY,Z//

αDF (X),F (Y ),F (Z)

F (X)⊗D F (Y ⊗C Z)JX,Y ⊗CZ // F (X ⊗C (Y ⊗C Z))

F (αCX,Y,Z)

(F (X)⊗D F (Y ))⊗D F (Z)JX,Y ⊗DidF (Z)

// F (X ⊗C Y )⊗D F (Z) JX⊗CY,Z

// F ((X ⊗C Y )⊗C Z).

F (1C)⊗D F (X)J1C,X //

J0⊗idF (X)

F (1C ⊗C X)

F (lCX)

1D ⊗D F (X)lDF (X)

// F (X),

F (X)⊗D F (1C)JX,1C //

idF (X)⊗DJ0

F (X ⊗C 1C)

F (rCX)

F (X)⊗D 1DrDF (X)

// F (X)

From now on, we will denote the functor C × C → C × C that permutes the factorsby Π. Notice that we can form the opposite monoidal category Cop of a monoidalcategory C = (C,⊗, α,1, l, r) by setting

Cop := (C,⊗ Π, α−1 Π,1, l, r).

Definition 5.2.0.3. Let C = (C,⊗, α,1, l, r) be a monoidal category. A symmetry ofC is an isomorphism of functors β : ⊗ → ⊗Π such that β(Π) β : ⊗ → ⊗ coincideswith the identity on ⊗ and for every X,Y, Z ∈ Ob(C), the diagram

(Y ⊗ Z)⊗XβY ⊗Z,X

((QQQQQ

QQQQQQ

QQ

(Z ⊗ Y )⊗X

βZ,Y ⊗idX66mmmmmmmmmmmmm

α−1Z,Y,X

X ⊗ (Y ⊗ Z)αX,Y,Z

Z ⊗ (Y ⊗X)

idZ⊗βY,X ((QQQQQ

QQQQQQ

QQ(X ⊗ Y )⊗ Z

Z ⊗ (X ⊗ Y )

βZ,X⊗Y

66mmmmmmmmmmmmm

commutes. This property is called the hexagon axiom. A symmetric monoidal cat-egory consists of the data C = (C,⊗, α,1, l, r; β) of a monoidal category C anda fixed symmetry β on C. In addition, if C = (C,⊗C, α

C,1C, lC, rC; βC) and D =

(D,⊗D, αD,1D, l

D, rD; βD) are symmetric monoidal categories, a symmetric monoidalfunctor C→ D is a monoidal functor (F, J, J0) such that for every X,Y ∈ Ob(C), thefollowing diagram commutes

F (X)⊗D F (Y )JX,Y //

βDF (X),F (Y )

F (X ⊗C Y )

F (βCX,Y )

F (Y )⊗D F (X)

JY,X // F (Y ⊗C X).

47

Remark 5.2.0.4. In many books of the literature, the construction of symmetricmonoidal categories is slightly different. For instance, in [Del] and [Mil3], we take acategory C together with a tensor product ⊗ : C2 → C, an associativity constraintϕX,Y,Z : X ⊗ (Y ⊗ Z) → (X ⊗ Y ) ⊗ Z, which is in fact ϕ = α from the definition, acommutativity constraint, which is a functorial isomorphism ψX,Y : X ⊗ Y → Y ⊗Xsuch that ψY,X ψX,Y : X ⊗ Y → X ⊗ Y which is clearly equivalent to the symmetryin 5.2.0.3. Instead of using it as a definition, ϕ and ψ are said to be compatible if theysatisfy the hexagon axiom. Finally, Deligne and Milne define an identity object of(C,⊗) as a pair (U, u) such that U ∈ Ob(C) is an object satisfying that X 7→ U⊗X isan equivalence of categories and u : U → U ⊗U is an isomorphism, in order to definea tensor category as a system (C,⊗, ϕ, ψ) where ϕ and ψ are compatible associativityand commutativity constraints respectively and there exists an identity objects. Inproposition 1.3 of [Mil3], we see how to construct a symmetric monoidal categoryusing the existence of an identity object.

5.3 Internal homomorphisms and rigidityThe goal of this section is to give a short introduction to internal homomorphisms andrigidity over symmetric monoidal categories, which allow us to endow such categorieswith the notion of a ”dual”.

Definition 5.3.0.1. Let C be a monoidal category, and let X,Y ∈ Ob(C). If

HomC(−⊗X,Y ) : Copp → Set

is representable, then we denote the representing object by Hom(X,Y ) and call it theinternal homomorphism of X and Y .

Now, assume that Hom(X,Y ) exists for a pair of objects X,Y ∈ Ob(C). Then, wehave a natural isomorphism

HomC(Hom(X,Y )⊗X,Y )∼−−−→ HomC(Hom(X,Y ),Hom(X,Y )),

given by the definition of internal homomorphism and what we call the evaluationmap, evX,Y : Hom(X,Y )⊗X → Y , which corresponds to idHom(X,Y ) via the previousisomorphism.. Hence, for every ϕ : T ⊗ X → Y , there exists a unique morphismψ : T → Hom(X,Y ) such that the following diagram commutes.

T ⊗X ϕ //

g⊗idX ((PPPPP

PPPPPP

P Y

Hom(X,Y )⊗XevX,Y

77pppppppppppp

In the next remark, we see a property of internal homomorphisms, that will help usfind isomorphisms between them.

48

Remark 5.3.0.2. Assume that internal homomorphisms exists for any pair of objectsin C. Let X,Y, Z ∈ Ob(C). Then, for every T ∈ Ob(C), we have the following sequenceof isomorphisms

HomC (T,Hom(Z,Hom(X,Y )) ≃ HomC(T ⊗Z,Hom(X,Y )) ≃ HomC((T ⊗Z)⊗X,Y )

and the latter is isomorphic to HomC(T,Hom(Z ⊗X,Y ), which leads us to the factthat

Hom(Z ⊗X,Y ) ≃ Hom(Z,Hom(X,Y ))

Definition 5.3.0.3. Let C be a monoidal category and let X be an object of C. Thedual of X is Hom(X,1) if it exists, and we denote it as X∨.If X∨ exists, we can rewrite the evaluation map as evX : X∨ ⊗X → 1.Remark 5.3.0.4. Assume that C is a symmetric category. Clearly, we have a mor-phism evX βX,X∨ : X⊗X∨ → 1, but since HomC(X⊗X∨) ≃ HomC(X, (X

∨)∨), theremust exist a unique map iX ∈ HomC(X, (X

∨)∨) that corresponds to evX βX,X∨ .The following definition gives sense to what we meant by the beginning of the sub-section when we talked about the notion of a dual.Definition 5.3.0.5. Let X ∈ Ob(C). If the map iX of the previous definition is anisomorphism, we say that X is reflexive.Of course, it is important to see if we can also endow the duals with a notion of “dualmap”. We solve the question in the following remark.Remark 5.3.0.6. Let X,Y ∈ Ob(C), and assume that X∨ and Y ∨ exist. Let f ∈HomC(X,Y ). Then, there exists a unique map f∨ : Y ∨ → X∨ such that the diagram

Y ∨ ⊗X f∨⊗idX //

idY ∨⊗f

X∨ ⊗XevX

Y ∨ ⊗ Y evY

// 1

commutes. It is also important to notice that if X,Y are reflexive, (f∨)∨ = f .Finally, we can construct the following natural maps, which will give us a bijectivecorrespondence between the tensor product of internal homomorphisms between theelements of a family and the internal homomorphism of the product of the elements.Let (Xi, Yi)1≤i≤n be a family of pairs of Xi, Yi ∈ Ob(C). Then,⊗

1≤i≤n

Hom(Xi, Yi)⊗⊗1≤i≤n

Xi ≃⊗1≤i≤n

Hom(Xi, Yi)⊗Xi →⊗1≤i≤n

Yi

where the first isomorphism follows from the associativity and commutativity con-straints and the second morphism is given by evX1,Y1⊗· · ·⊗evXn,Yn , and this previousmap corresponds to ⊗

1≤i≤n

Hom(Xi, Yi)→ Hom(⊗1≤i≤n

Xi,⊗1≤i≤n

Yi).

Thus, we can give the following definition.

49

Definition 5.3.0.7. Let C be a symmetric monoidal category. C is rigid if it satisfies

1. For any X,Y ∈ Ob(C), the internal homomorphism Hom(X,Y ) exists.

2. Every X ∈ Ob(C) is reflexive.

3. For any family (Xi, Yi)1≤i≤n, where Xi, Yi ∈ Ob(C), the map⊗1≤i≤n

Hom(Xi, Yi)→ Hom(⊗1≤i≤n

Xi,⊗1≤i≤n

Yi)

is an isomorphism.

And from the definitions it follows immediately that

Proposition 5.3.0.8. Let C be a rigid symmetric monoidal category. Then, for anyX1, . . . , Xn, X, Y ∈ Ob(C),

1. X∨ ⊗ Y ≃ Hom(X,Y ), and

2.⊗

1≤i≤nX∨i ≃

(⊗1≤i≤nXi

)∨.

Proof. For the first point, all we have to do is apply the third point in 5.3.0.7, thatis,

X∨ ⊗ Y ≃ Hom(X,1)⊗ Hom(1, Y )∼−→ Hom(X,Y ).

For the second point, we apply the same isomorphism,

⊗1≤i≤n

X∨i =

⊗1≤i≤n

Hom(Xi,1) ≃ Hom(⊗

1≤i≤n

Xi,1

)=

(⊗1≤i≤n

Xi

)∨

.

One important property of rigid symmetric monoidal categories is the following. Ac-tually, we need it in order to show Theorem 6.2.0.7.

Proposition 5.3.0.9. Let C and D be rigid symmetric monoidal categories, and letF,G : C → D be monoidal functors. Then, every morphism Φ : F → G is anisomorphism of functors.

Proof. Proposition 1.13 of [Mil3].

In the end of the next chapter we see that the following definition is a particular caseof a more general definition, although this one is the adequate for the case of affinealgebraic groups.

Definition 5.3.0.10. Let C = (C,⊗, α,1, l, r; β) be a symmetric abelian monoidalcategory. A fiber functor is an exact and faithful monoidal functor (F, J, J0) : C →fVeck, where fVeck denotes the category of finite dimensional k-vector spaces. Also,a Tannakian category is an abelian, rigid, symmetric monoidal category admitting afiber functor. Furthermore, we say that a tannakian category is neutral if its unit 1satisfies End(1) ≃ k.

50

We conclude this section with another important definition. The reason why thisdefinition is relevant is because in many cases we can reduce the study of a givenTannakian category C to the study of its full subcategories “tensor generated” by agiven element.

Definition 5.3.0.11. Let C be a Tannakian category and let X ∈ Ob(C). Let ⟨X⟩⊗be the full subcategory whose objects are the quotients of elements of the form(⊕

i∈I

X⊗Cni ⊗C (X∨)⊗Cmi

).

We call ⟨X⟩⊗ the category tensor generated or tensor subgenerated by X.

In fact, in Corollary 6.20 of [Del], Deligne shows the following result.

Proposition 5.3.0.12. If an abelian, rigid, k-linear symmetric monoidal category Cadmits full subcategory tensor generated by an element of C and the latter admits afibre functor, then C admits a fibre functor over a finite convenient extension K of k.

5.4 Examples of tannakian categoriesLet us see a few examples of Tannakian Categories. Here, we see that all of them areequivalent (in the categorical sense) to the category of representations of an affinealgebraic groups, which leads us to a natural question: Are all Tannakian categoriesequivalent to Rep(G) for a certain affine algebraic group G? In the next chapter wesee that the answer is yes.

Proposition 5.4.0.1. The category of linear representations of an affine algebraicgroup G, Rep(G) is a Tannakian Category.

Proof. From 4.1.0.3, we have that Rep(G) forms a category. Also, we obtain that thetensor product is ⊗ := ⊗G, hence we can define α, l and r easily. From this followsthat Rep(G) is a monoidal category. On the other hand, we also saw in 4.1.0.3 how tobuild, for anyM,N ∈ Ob(Rep(G)), a canonical isomorphismM⊗GN → N ⊗GM,which defines the symmetry of Rep(G), therefore we have that it is a symmetricmonoidal category.

Now let us see that it is rigid. All we have to do is notice that for anyM1,M2,M3 ∈Ob(Rep(G)), we have a canonical isomorphism

HomRep(G)(M1 ⊗GM2,M3) ≃ HomRep(G)(M1,Hom(M2,M3)),

hence the functor HomRep(G)(− ⊗GM,N ) : Rep(G)opp → Set is represented byHom(M,N ) =: Hom(M,N ), so we have the existence of internal homomorphismsfor each M,N ∈ Ob(Rep(G)). Clearly, all M ∈ Ob(Rep(G)) are reflexive, becauseof the natural isomorphism

Hom(Hom(X,1),1) ≃ X.

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Thus, the fact that Rep(G) is rigid follows from the fact that for a family (Mi,Ni)0≤i≤nof pairs of Mi,Ni ∈ Rep(G), evM1,N1 ⊗ · · · ⊗ evMn,Nn always gives an isomorphismbetween ⊗

1≤i≤n

Hom(Mi,Ni)⊗Mi →⊗1≤i≤n

Ni,

thus ⊗1≤i≤n

Hom(Mi,Ni) ≃ Hom(⊗

1≤i≤n

Mi,⊗1≤i≤n

Ni

).

Eventually, in order to find the fiber functor, we take the forgetful functor F :Rep(G)→ k −Mod, and notice that we can give it a structure of monoidal functorby setting JM,N := idM⊗N :M⊗N →M⊗N and J0 := idk : k → k. Clearly, it issymmetric, exact and faithful.

Now let’s see two other interesting examples. To start, let GradVeck denote thecategory of graded vector spaces. Recall that any object in this category is a familyof k-vector spaces (V n)n∈Z such that the family has a finite dimensional sum V :=⊕

n∈Z Vn.

Proposition 5.4.0.2. The category of graded vector spaces GradVeck is a Tannakiancategory.

Proof. All we have to do is define the tensor product ⊗ : GradVeck×GradVeck →GradVeck by taking (V n)n, (W

n)n ∈ Ob(GradVeck) and setting

(V n)n ⊗ (W n)n :=

(⊕i∈Z

V i ⊗k W n−i

).

We can set the unit to be 1 := (kδn,0)n ∈ Ob(GradVeck). Clearly, this is a symmetricmonoidal category. On the other hand, the internal homomorphisms are

Hom((V n)n, (Wn)n) =

(⊕i∈Z

Hom(V i,W i+n)

)n

,

and if we set the fiber functor to be the forgetful functor F : GradVeck → k−Modgiven by (V n)n 7→ V , the rest of points are clear.

Remark 5.4.0.3. Furthermore, it is interesting to notice that this category is actuallyequivalent to the category of representations of Gm. All we have to do is notice thateach (V n)n corresponds to the character χ that sends each λ to λn.

One last example is that of Hodge structures. If we let k = R, then a real Hodgestructure is a finite dimensional R-vector space V such that it admits a decomposition

V ⊗R C =⊕p,q

V p,q,

52

where V p,q and V q,p are complex conjugate subspaces of V ⊗R C, V p,q = V q,p. If welet the objects of the category of Hodge structures HSR be the real Hodge structures,then we notice that HSR is a Tannakian category, together with the fiber functor Fgiven by F : (V, V p,q) → V . The proof of this is left to the reader, as it is easy tocheck, that can be done by following the same steps as in the previous proposition.

Remark 5.4.0.4. It is easy to see that the category of real Hodge structures HSR isequivalent to the category of representations of the real algebraic group, that is, thealgebraic group S represented by the C-algebra C[X, 1/X], because any real Hodgestructure (V, (V p,q)) corresponds to the representation of S on V , by setting thatλ ∈ S(R) = C× acts on each V p,q by λ−pλ−q.

53

54

Chapter 6

Tannaka-Krein duality

In this chapter we prove Tannaka-Krein’s duality theorem and then we see thatany tannakian category is equivalent to the category of representations of a certainalgebraic group. These two results are the cornerstone of the memoir, because theyallow us to relate all the theory we have built up to this point to differential Galoistheory, among other branches of mathematics.

6.1 The tannakian reconstructionLet G be an affine algebraic group. We know that the tannakian category Rep(G)comes with a fiber functor ω : Rep(G) → fVeck, that sends M 7→ M , hence wecan extend it, for each A ∈ Ob(Algk), to a functor ωA : Rep(G)→ModA given byM 7→M ⊗k A. Let us see that this functor has good properties.

Lemma 6.1.0.1. For each A ∈ Ob(Algk), ωA is a monoidal functor.

Proof. All we have to do is notice that ModA is a symmetric monoidal categorytogether with the tensor product ⊗ := ⊗A. The diagrams from the definition ofmonoidal functor commute thanks to the properties of Rep(G) and ModA.

Clearly, ωA induces a functor Aut⊗(ω) : Algk → Grp, defined by

Aut⊗(ω)(A) := (λM)M∈Ob(Rep(G)) : λM ∈ AutA(M ⊗k A),

∀α ∈ HomRep(G)(M,N ), (α(k)⊗ idA)λM = λN (α(k)⊗ idA).Notice as well that if we take g ∈ G(A), we can obtain g ∈ Aut⊗(ω)(A) by setting, foreach M∈ Rep(G), gM to be the action of g on WM(A), which gives us a functorialrelationship, hence G→ Aut⊗(ω) is a functor homomorphism.Our goal now is to show that G→ Aut⊗(ω) is an isomorphism. In order to do so, weneed the following lemma.

Lemma 6.1.0.2. Let V be a k-vector space and let G := GLV . Also, let I ⊂ Γ(G,O)be a Hopf ideal such that the k-algebra Γ(G,O)/I defines an algebraic group Hembedded in G. Then, there exists a finite dimensional k-vector space W and a lineD ⊂ W such that H is the stabilizer of D on W .

55

Proof. Let F ⊂ Γ(G,O) be a k-vector subspace of Γ(G,O) generated by a finitesystem of generators of I. Let ρ : G×G→ G be the action of G on itself that sendseach pair (x, y) to yx−1. By following 3.4.2 a) in [Cre], we obtain a finite dimensionalk-subspace W of Γ(G,O) containing F such that is stable through all ρx : Γ(G,O)→Γ(G,O), given by ρx(g)(y) = g(yx). Let M := W ∩ I so that M generates I. Noticethat M is stable under all ρx, for all x ∈ H since H = x ∈ G : ρxI = I (see [Cre]Lemma 3.4.1). If ρgM =M , since M generates I, ρxI = I so x ∈ H. If we take nowL = ∧dimMM , since : ρxM =M if and only if ρxL = L, by Lemma 3.7.1 of [Cre], wehave the characterization of H we wanted.

Theorem 6.1.0.3. The functor homomorphism G→ Aut⊗(ω) is an isomorphism.

Proof. Let M ∈ Ob(Rep(G)) be a linear representation of G and ⟨M⟩⊗ be the fullsubcategory of Rep(G) that is tensor subgenerated by M. Also, let , ω|⟨M⟩⊗ be therestriction of ω to ⟨M⟩⊗. In addition, we may also refer to Aut(ω|⟨M⟩⊗) as a k-groupfunctor. Notice that we have an embedding

Aut⊗(ω|⟨M⟩⊗)→ GLM = Aut⊗(WM).

Where WM(A) = A ⊗M , because of the map λ = (λN )N∈Ob(Rep(G)) 7→ λM, whichidentifies Aut⊗(ω|⟨M⟩⊗)(A) with a subgroup of GLM . On the other hand, denote byGM the image of G in GLM through the previous morphism, which is an algebraicsubgroup of GLM that satisfies

GM(A) ⊆ Aut⊗(ω|⟨M⟩⊗)(A) ⊆ GLM(A).

Our goal now is to see that the image of Aut⊗(ω|⟨M⟩⊗) in GLX coincides with GM.Thanks to the previous lemma it is enough to check that Aut⊗(ω|⟨M⟩⊗) leaves invari-ant every vector that is invariant under GM. Let N ∈ Ob(⟨M⟩⊗) and let v ∈ Vbe invariant under GM. In particular, the map ρ : triv → N such that 1 7→ v isG-equiinvariant. Also, since ω is a tensor functor, for every automorphism λ of Fand every A ∈ Ob(Algk),

λV (A)(v ⊗ 1A) = ρλtriv(A)(1⊗ 1A) = v ⊗ 1A,

so we have that GM = Aut⊗(ω|⟨M⟩⊗). Now, if M2 is a subrepresentation of M1, wehave a commutative diagram

GM1//

Aut⊗(ω|⟨M1⟩⊗)

GM2

// Aut⊗(ω|⟨M2⟩⊗),

where the vertical maps are given by the restriction. Since the regular representationof G is faithful, we have that

G = lim←−G∈Ob(Rep(G))

GM, Aut⊗(ω) = lim←−M∈Ob(Rep(G))

Aut⊗(ω|⟨M⟩⊗).

where the right equality is clear. Hence, G ≃ Aut⊗(ω).

56

Before ending this subsection, it would be interesting to see that if we take two affinealgebraic groups G and G′, under certain hypothesis we can find a ”good” relation-ship between both of them. More specifically, notice that if we take an algebraic k-homomorphism f : G→ G′, we can define a tensor functor ωf : Rep(G′)→ Rep(G)such that ωG ωf = ωG

′ , using M = (M,ΦM) 7→ (M,ΦM f). Let us go a little bitfurther.

Corollary 6.1.0.4. Let G and G′ be affine algebraic k-groups and F : Rep(G′) →Rep(G) a monoidal functor such that ωG F = ωG

′. Then, there exists a uniquealgebraic k-homomorphism f : G→ G′ such that F = ωf .

Proof. We know that F induces an homomorphism F ∗ : Aut⊗(ωG)(R)→ Aut⊗(ωG′)(R)

via F ∗(λM) = λF (M). Thanks to 6.1.0.3 and Yoneda’s lemma, we can identify F ∗

with an algebraic k-homomorphism f : G → G′, where F 7→ F ∗ and f 7→ ωf areclearly the inverse maps.

6.2 Tannakian Categories and Algebraic GroupsOur goal now is to show that if C is an abelian, rigid, symmetric monoidal k-linearcategory and ω : C→ fVeck is a fiber functor for the category,

1. The functor Aut⊗(ω) is an algebraic group.

2. The functor C→ Rep(Aut⊗(ω)) induced by F is an equivalence of categories.

In order to proceed with the proof, we need a few previous results. For the firstproposition, remember that a skeleton for a category C is an equivalent category Din which no two distinct objects are isomorphic. A more precise definition can befound in [Ma]. Also, from now on, C will denote an abelian rigid symmetric monoidalk-linear category.

Proposition 6.2.0.1. There exists a functor ⊠ : Modk ×C→ C such that for everyX,Y ∈ Ob(C) and V ∈ Ob(fVeck),

1. HomC(X,V ⊠Y ) ≃ V ⊗kHomC(X,Y ) and HomC(V ⊠Y,X) ≃ V ⊗kHomC(Y,X).

2. For any k-linear functor F : C→ fVeck, F (V ⊠X) ≃ V ⊗k F (X).

Proof. First of all, we pick the standard skeleton of fVeck, fVecsk. Remember thatthe objects of fVecsk are the V of the form kn for a certain n ∈ N ∪ 0. Also, forevery finite dimensional k-module V ∈ Ob(fVec) we can define an isomorphism ϕV :kdim(V ) → V . Therefore, following [Ma] there exists a unique equivalence of categoriesΓ : fVeck → fVecsk such that its quasi-inverse is the inclusion i : Modsk → fVeck.Because of the latter observation, we have a natural isomorphism δ : Γ i → idfVec

given by the existence of the quasiinverse. Now we can define ⊠ easily - for eachn ∈ N ∪ 0 and X ∈ Ob(C), kn ⊠ X := X⊕n , and for each V ∈ Ob(fVeck),V ⊠ X := Γ(V ) ⊠ X. Thus, all that we have to do is verify that the isomorphisms

57

hold. The first one follows directly from the fact that there exists an n ∈ N ∪ 0such that

HomC(X,V ⊠ Y ) ≃ HomC(X,Γ(V )⊠ Y ) = HomC(X,Y⊕n

) ≃ kn ⊗k HomC(X,Y ),

and for the second point, F (V ⊠X) = F (X⊕n) = kn ⊗k F (X). Hence, we are done

with the proof.

Now, for a k-module V and an object X ∈ Ob(C), we define Hom(V,X) := V ∨ ⊠X.In some books of the literature, for instance [Mil3], Hom is written as Hom, butwe prefer to write it in the first way in order to avoid confusion with the internalhomomorphisms. An important question that raises naturally is: how can we definea subspace of Hom(V,X) for any k-submodule W of V and any subobject Y ⊆ X?.The following definition answers the previous question.

Definition 6.2.0.2. Taking the previous notations, we define the transporter of Wto Y as

(Y : W ) := ker(Hom(V,X)→ Hom(W,X/Y )).

Notice that if F : C → Modk is an exact k-linear functor, thanks to 6.2.0.1,F (Hom(V,X)) = F (V ∨ ⊠X) ≃ HomModk

(V, F (X)), hence

F (Y : W ) = (F (X) : W ) = f ∈ HomModk(V, F (X)) : f(W ) ⊆ F (Y ).

In order to show the result of the beginning of the subsection, we will follow thenext plan. Just like we saw in 6.1.0.3, we want to restrict C to a subcategory tensorgenerated by an element X ∈ Ob(C), ⟨X⟩⊗. More precisely, we want to show that⟨X⟩⊗ is equivalent to the category of comodules over a coalgebra. Using inverse limits,we will see that the entire category C is equivalent to the category of comodules overanother coalgebra, which turns out to be a Hopf algebra.In order to see that ⟨X⟩⊗ is equivalent to the category of comodules over a coalgebraC, denoted as CoModC , we see first that ω| ˜⟨X⟩⊗ identifies CX with CoModC . Noticethat if C is finite dimensional, then there exists an equivalence of categories

CoModC∼−−−→ModC∨ .

Hence, all we have to do is see that ⟨X⟩⊗ is equivalent to the category of modulesover a certain algebra R.

Lemma 6.2.0.3. For any X ∈ Ob(C), the following two subobjects of Hom(ω(X), X)are equal.

1. The largest subobject P ⊆ Hom(ω(X), X) whose image in Hom(ω(X)⊕n, X⊕n)is contained in (Y : ω(Y )) for any Y ⊆ X⊕n and n ∈ N.

2. The smallest subobject P ′ ⊆ Hom(ω(X), X) such that ω(P ′) ⊆ ω(Hom(ω(X), X)) =HomModk

(ω(X), ω(X)) contains idω(X).

58

Proof. First, notice that the existence of the fiber functor ω : C→ fVeck implies thatC is artinian and noetherian. Therefore, both P and P ′ are well-defined subobjectsof Hom(ω(X), X). Now, by definition,

P =∩n≥0

∩Y⊆X⊕n

Hom(ω(X), X) ∩ (Y : ω(Y )),

ω(P ) =∩n≥0

∩Y⊆X⊕n

EndfVeck(ω(X)) ∩ (F (Y ) : F (Y )).

This shows that ω(P ) is the largest subring of EndfVeck(ω(X)) stabilizing all F (Y ),Y ⊆ X⊕n. Hence, idX ∈ ω(P ) and P ′ ⊆ P . For the reverse inclusion, notice thatif we take a subobject Y ⊆ Hom(ω(X), X), by definition of P , left multiplication byω(P ) ⊆ EndfVeck(ω(X)) stabilizes ω(Y ) ⊆ EndfVeck(ω(X)). Hence, that fact thatidω(X) ∈ ω(P ′) implies that ω(P ) ⊆ ω(P ′), that is, P ⊆ P ′.

After all these preliminaires, we can continue with the construction of a coalgebra.Let X ∈ Ob(C) and let PX ⊆ Hom(ω(X), X) be the object defined in 6.2.0.3. Also,let ⟨X⟩ ⊆ C be the subcategory whose objects are subobjects of quotients of X⊕n .By definition, the functor ω|⟨X⟩ : ⟨X⟩ → fVeck factors through fVecω(PX). From nowon, let us denote AX := ω(PX). Then,

Proposition 6.2.0.4. For any Y ∈ Ob(⟨X⟩), there is a natural action of AX onω(Y ). Furthermore, ω|⟨X⟩ : ⟨X⟩ → fVecAX

is an equivalence of categories sendingω|⟨X⟩ to the forgetful functor, and AX = End(ω|⟨X⟩).

Proof. The first point follows from applying the second point of 6.2.0.1 with F = ω.From that, we obtain AX = ω(PX) ⊂ ω(X)∨ ⊗ ω(X), which is isomorphic toHomfVeck(ω(X), ω(Y )) and therefore AX operates on ω(X), and this action is natu-rally extended to ω(Y ) for any Y ∈ Ob(⟨X⟩). On the other hand, notice that we havean action of AX ⊆ End(ω(X)∨) on Hom(ω(X), X) and it is clear that this action sta-bilizes PX . Now, if M is a right A-module, then we get two maps ((M⊗AX)⊠PX)→M ⊠PX , one by considering the action of AX on M and the other by considering theaction on PX . We define M ⊗AX

PX to be the equalizer of these maps. By definition,M ⊗AX

PX ∈ Ob(⟨X⟩) so we have ω(M ⊗AXPX) =M ⊗AX

ω(PX) =M , that is, ω isessentially surjective. Now, if f :M → N is an AX-module map, then we may definea map

M ⊗AXPX → N ⊗AX

PX

that shows that ω is full. Finally, the faithfulness of ω follows by hypothesis. Hence,we have shown that ω is a category equivalence. The last point is clear.

So now, we can let CX := A∨X , so that ⟨X⟩ is equivalent to the category of C-

comodules. Thus, we have

Corollary 6.2.0.5. Let H := lim−→End(F |⟨X⟩)∨. Then, ω factors through CoModH

and moreover, it is an equivalence between C and the category of CoModH sendingω into the forgetful functor.

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Now, our goal is to define a commutative Hopf algebra structure on H. We will usethe fact that C is symmetric. In order to do so, let B ∈ Ob(CoAlgk) and considerF : CoModB → fVeck. It is clear that

B = lim−→X∈Ob(C)

End(F |⟨X⟩)∨.

If we also observe that for a finite dimensional algebra A, there is an isomorphismA ≃ End(FA), we have that any functor CoModB → CoModB′ that carries F toitself arises from a unique coalgebra homomorphism B → B′.The last step before we conclude the proof is the following lemma. Notice that if wehave a coalgebra homomorphism B ⊗k B → B, we can define a functor CoModB ×CoModB → ComodB via (X,Y ) 7→ X ⊗k Y , with comodule structure defined bythe the coalgebra homomorphism.

Lemma 6.2.0.6. The previously defined map defines a bijective correspondence be-tween the set of coalgebra homomorphisms of the form B ⊗k B → B and the setof functors of the form CoModB × CoModB → ComodB. The product inducedby the coalgebra homomorphism is associative (resp. commutative) if and only ifthe natural associativity (resp. commutativity) constraint on fVeck indues a similarconstraint on CoModB. The product induced by that same morphism has a unit ifand only if CoModB has a unit object with underlying k-module k.

So we can finally show the main result of this section.

Theorem 6.2.0.7. Let C be an abelian, rigid, symmetric monoidal k-linear categoryand ω : C→ fVeck a fiber functor for C. Then,

1. The functor Aut⊗(ω) is an algebraic group, called the tannakian fundamentalgroup of (C, ω).

2. The functor C→ Rep(Aut⊗(ω)) induced by F is an equivalence of categories.

Proof. Taking the notations that we have followed throughout this subsection, 6.2.0.6shows that H := lim−→End(ω|⟨X⟩)

∨ is a commutative algebra with identity. Noticethat G := Spm(H) is an affine algebraic monoid. Just like in 6.1, we have thatG ≃ End⊗(ω). Since both C and fVeck are rigid, proposition 5.3.0.9 shows thatEnd⊗(ω) = Aut⊗(ω), so G is in fact an affine algebraic group and thanks to 3.2.0.2,H is a Hopf algebra.

6.3 A note on fibre functorsWe begin with the definition we mentioned before 5.3.0.10.

Definition 6.3.0.1. Let C be a symmetric monoidal category and let X be an alge-braic k-scheme. A fibre functor of C on S is an exact, faithful and k-linear symmetricmonoidal functor

ω : C→ QCoh(OS),where OS is the sheaf associated to the algebraic scheme S.

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From the definition of a symmetric monoidal category and the properties of the pre-viously defined functor follows that ω is in fact a functor from C to the category oflocally free sheaves of finite rank on OS. In fact, since the case that concerns us themost is affine algebraic schemes, notice that if S = Spm(A), then ω can be identifiedwith a functor C → fProjModA, where fProjModA denotes the category of finiteprojective modules over A. Notice that fProjodA is a symmetric monoidal category,and this can be shown taking into account the fact that ModA is also a symmetricmonoidal category.

Definition 6.3.0.2. If S is an affine algebraic k-scheme, then the previous functorω : C→ fProjModΓ(S,OS)

is called a fibre functor on Γ(S,OS).

So let us start giving a proposition which, after a few observations, shows us thatDefinition 6.3.0.2 is equivalent to 5.3.0.10.

Definition 6.3.0.3. Let A be a k-algebra and let X ∈ Ob(ModA). If X∨ exists,then X is a projective A-module of finite rank.

Proof. Let Y ∈ Ob(ModA). Thanks to 5.3.0.8, we have that X∨ ⊗ Y ≃ Hom(X,Y ).Thanks to the properties of ModA, we have that Hom(X,Y ) = HomModA

(X,Y ).From this, if we let Y = A, then we have that X∨ is the dual of X. Now, let Y = X.Using the previous isomorphism, we have an isomorphism

X∨ ⊗X → HomModA(X,X).

If the image throught the previous isomorphism of∑αk⊗xk is idX , then the αk and

xk define a factorisation of the identity

Xα−→ An

x−→ X.

Hence, X is a direct factor of a free module of finite rank.

Notice that if we are given a symmetric monoidal functor F : C→ D and X ∈ Ob(C)admits a dual X∨, F (X∨) is a dual of F (X) thanks to 5.3.0.9. Hence, let V be a finitedimensional k-vector space and let X be an object of an additive k-linear categoryA. Thanks to 6.2.0.1, we have the existence of a tensor product ⊠ that satisfies

Hom(Y, V ⊠X) = V ⊗k Hom(Y,X).

If we take a look at the proof of 6.2.0.1, we can identify V ⊗X withXdim(V ). Hence, thefull-subcategory of C formed by the multiples 1n of 1 is equivalent to the categoryof finite dimensional k-vector spaces via the morphism V 7→ V ⊠ 1. Thanks toProposition 1.17 of [Mil3], this equivalence is stable for subquotients, and thereforewe have shown the equivalence between 6.3.0.2 and 5.3.0.10.Before we conclude this section, it is also important to mention a generalization ofthe affine algebraic group Aut⊗(ω) that we introduced in the beginning of 6.1.

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Remark 6.3.0.4. In this case, let S denote an algebraic k-scheme and let ω1, ω2 : C→QCoh(OS) be two fiber functors on S. In [Del], Deligne defines an ⊗-isomorphismas a natural transformation u : ω1 → ω2 that makes the following diagram commute

ω1(X)⊗ ω1(Y ) ∼ //

u⊗u

ω1(X ⊗ Y )

u

ω2(X)⊗ ω2(Y ) ∼

// ω2(X ⊗ Y )

and u(1) : ω1(1) → ω2(1) is the identity isomorphism of OS, and later definesIsom⊗

S (ω1, ω2) as the functor that adjoints to any u : T → S all the fibre functor⊗-isomorphisms from u∗ω1 to u∗ω2, where u∗ is the dual of u : T → S, namelyu∗ : QCoh(OS)→ QCoh(OT ) which allows us to define

Isom⊗k (ω1, ω2) := Isom⊗

S1×S2(π∗

1ω1, π∗2ω2),

Aut⊗k (ω) := Isom⊗k (ω, ω).

where πi are the natural projections and π∗i ωi : C → QCoh(OSi

), because ωi : C →QCoh(OS1×S2). This allows Deligne to show that, by taking the definitions in point1.5 of [Del], if ω is a fibre functor on a non-empty algebraic k-scheme S of a k-tensorcategory C,

1. ω induces an equivalence between C and the category of representations of Aut⊗k ,Rep(S : Aut⊗k (ω)).

2. In particular, there is an isomorphism G∼−→ Aut⊗k (ω).

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Chapter 7

Application: Differential GaloisTheory

7.1 MotivationWe are about to study three important examples of tannakian categories: the categoryof complex local systems on a connected and locally simply connected topologicalspace X, denoted as LSX , the category of holomorphic connections on a connectedopen set D ⊆ C, ConnD, and the category of differential modules over the fieldC(t), DiffModC(t). In fact, we will see that there is an equivalence between the threecategories.

7.1.1 LSX and ConnD

Let us start with the following definition.

Definition 7.1.1.1. Let X be a connected and locally simply connected topologicalspace. A complex local system on X is a locally constant sheaf of finite dimensionalcomplex vector spaces.

It is clear that if we endow it with the usual tensor product and dual space, we seethat it is a C-linear abelian, rigid symmetric monoidal category. Now let us constructthe category of holomorphic connections.

Definition 7.1.1.2. A holomorphic connection on a connected open set D ⊆ C isa pair (E ,∇), where E is a locally free sheaf on D and a connection map ∇ : E →E ⊗O Ω1(D) is a morphism of sheaves of C-vector spaces satisfying

∇(fs) = df ⊗ s+ f∇(s)

for all open subsets U ⊆ D, f ∈ Γ(U,O) and s ∈ Γ(U, E), where Ω1(D) is the sheafof 1-forms over D, d is the usual derivation in C and the tensor product E ⊗O Ω1(D)is defined by the rule

U 7→ Γ(U, E)⊗Γ(U,O) Γ(U,Ω1(D)).

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Also, a morphism between (E ,∇) and (E ′,∇′) is a morphism of O-modules ϕ : E → Emaking the following diagram commute

E ∇ //

ϕ

E ⊗O Ω1(D)

ϕ⊗id

E ′∇′

// E ′ ⊗O Ω1(D).

Furthermore, given a section s ∈ Γ(U, E) of a connection (E ,∇), we say that s ishorizontal if it satisfies ∇(s) = 0. Horizontal sections form a subsheaf of C-vectorspaces E∇ ⊆ E .

Clearly, we can define the category of holomorphic connections on D, ConnD. Let usdefine a tensor product of connections. In fact, we are going to see that we can endowit with a neutral tannakian category structure. Let (E ,∇), (E ′,∇′) ∈ Ob(ConnD).We define the tensor product of (E ,∇) and (E ′,∇′) as (E⊗OE ′,∇⊗∇′) ∈ Ob(ConnD)where ∇⊗∇′ : E ⊗O E ′ → E ⊗O E ′ ⊗O Ω1(D) is given by

(∇⊗∇′)(s⊗ s′) = ∇(s)⊗ s′ + s⊗∇′(s′).

This gives ConnD the structure of a symmetric monoidal category. Let us see that itis rigid. All we have to do is define the dual connection to (E ,∇), (E∗,∇∗), where E∗is the locally free sheaf given by U 7→ Hom(E|U ,O|U) and ∇∗ over ϕ ∈ Hom(E|U ,O|U)via

∇∗(ϕ)(s) = 1⊗ dϕ(s)− (ϕ⊗ idΩ1(D))(∇(s)),so all that we need is a fiber functor, but we can define it naturally via

ωConnD : ConnD →Modk

(E ,∇) 7−→ E∇x ,

where x ∈ D is a previously fixed point. Finally, we see that ConnD is a C-linear,abelian category, because in [Sza] (Prop. 2.7.5) it is shown that there is and equiv-alence between ConnD and LSD, compatible with the monoidal category structureand fiber functors, where the fiber functor of LSX , for a connected and locally simplyconnected topological space X is given by taking the stalk of a local system at a fixedpoint x ∈ X, which shows that LSX is a neutral tannakian category. With this, wehave shown that

Proposition 7.1.1.3. The category of holomorphic connections ConnD endowedwith the previous tensor products, duals and fiber functor is a tannakian category.

7.1.2 DiffModC(t)

Now, let (C, ddt) denote the pair formed by the field C(t) together with its usual

derivation ddt

. Although we define this notion in a more general way in the nextsection, a differential module over C(t) is a pair (V,∇), where V ∈ Ob(VecC(t))

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and ∇ : V → V is a C-linear map satisfying ∇(fv) = dfdtv + f∇v. Clearly, given two

differential modules over C(t), we can define a morphism between them as a morphismof C(t)-vector spaces that is compatible with ∇, which gives us the arrows in thecategory DiffModC(t), whose objects are the differential modules. This category isclearly abelian, because we can define differential submodules as C(t)-subspaces of agiven differential module (V,∇) that are also compatible with ∇. Next, we endow thecategory with a tensor product. Let (V,∇), (V ′,∇′) ∈ Ob(DiffModC(t)). We denoteV ⊗ V ′ as the usual tensor product of C(t)-vector spaces and ∇⊗∇′ by

(∇⊗∇′)(s⊗ s′) := ∇(s)⊗ s′ + s⊗∇′(s′), ∀s⊗ s′ ∈ V ⊗ V ′,

and ∇∗ on V ∗ = HomVecC(t)(V,C(t)) by the rule ∇∗(ϕ)(v) = ddtϕ(v)− ϕ(∇(v)).

Proposition 7.1.2.1. The category DiffModC(t) together with the previous tensorproduct and duals, is a C-linear rigid tensor abelian categoryNow let us see how we can relate the category of differential modules to the categoryof connections. Notice that if we take an n-dimensional C(t)-vector space V andidentify it with C(t)n, we can define a derivation D,

D(f1, . . . , fn) :=

(df1dt, . . . ,

dfndt

), ∀f1, · · · , fn ∈ C(t)

so that (V, d) is a differential module. Notice that thanks to the definition of adifferential module, more precisely, the definition of ∇, we have that ∀f, f1, . . . , fn ∈C(t),

(∇−D)(f(f1, . . . , fn)) =df

dt(f1, . . . , fn) + f∇(f1, . . . , fn)−

−(df

dtf1 + f

df1dt, . . . ,

df

dtfn + f

dfndt

)= f(∇−D)(f1, . . . , fn),

and (∇−D)((f1, . . . , fn)+ (g1, . . . , gn)) = (∇−D)(f1, . . . , fn)+ (∇−D)(f1, . . . , fn),hence ∇−D is C(t)-linear and therefore it has an associated matrix C ∈Mn×n(C(t))called the connection matrix of ∇. This is the last step before we show the followingresult.Proposition 7.1.2.2. DiffModC(t) endowed with the operations from 7.1.2.1 is atannakian category.Proof. For this proof, we need Prop. 2.7.7 of [Sza]. Let D ⊂ C be an open connectedand locally simply connected set such that its complement is finite. Our initial goalis showing that there is a categoric equivalence between the full subcategory tensorgenerated by a connection (E ,∇) of ConnD and the full subcategory tensor generatedby (V,∇), a differential C(t)-module The first result listed above states that for anholomorphic connection (E ,∇) ∈ Ob(ConnD), there is a connection (E ,∇) on P1

C withsimple poles outside D. This shows that the elements of the connection matrix of ∇are elements of C(t), so we can associate the differential module (V,∇) ∈ DiffModC(t)to (E ,∇) with the very same connection matrix. This leads to our initial goal, andalso shows the proposition, because the fact that there is a fiber functor on ConnDand the latter equivalence of subcategories shows that we can endow DiffModC(t)with a fiber functor.

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7.2 Differential Galois theoryWe give a quick introduction to some of the basic tools that give rise to what is knownas differential Galois theory. Afterwards, we will relate all those tools to the theoryof neutral tannakian categories, and see how it turns out to be a very powerful toolthat eventually leads us to the proof of the main theorem of the theory of differentialGalois theory.

7.2.1 Basic definitionsFrom this point to the end of this chapter, we assume that every field is of character-istic 0. Most of the notions we introduce here can be defined in a more general way.For that reason, we recommend Part 3 of [Cre].

Definition 7.2.1.1. Let K be a field. A derivation on K is an additive map ∂ : K →K such that the Leibniz rule holds,

∂(fg) = ∂(f)g + f∂(g), ∀f, g ∈ K.

We call the pair (K, ∂) a differential field. Let (K, ∂) and (K ′, ∂′) be differential fields.A differential morphism is a field homomorphism ϕ : K → K ′ such that

∂′(ϕ(f)) = ϕ(∂(f)), ∀f ∈ K

Also, an extension of differential fields (L, ∂′)|(K, ∂) is a field extension L|K suchthat ∂′ extends ∂.

The most simple example we have of a differential field is given by the trivial deriva-tion, given by ∂(f) = 0, for all f ∈ K. On the other hand, if (K, ∂) is a differentialfield, the kernel of ∂, which is a subfield of K, is called the constant field of (K, ∂). Itis commonly denoted as k. Now let us generalize the definition of differential modulewe give in the previous section.

Definition 7.2.1.2. Let (K, ∂) be a differential field. A differential module over(K, ∂) is a pair (V,∇) where V is a K-module and ∇ is a k-linear map ∇ : V → Vsatisfying

∇(fv) = ∂(f)v + f∇(v), ∀f ∈ K, v ∈ V.Also, the elements of V ∇ := Ker(∇) form a k-subspace of V and are called horizontalvectors.

Corollary 7.2.1.3. The category DiffModK of differential modules over a differen-tial field (K, ∂) is a tannakian category together with the fibre functor given by theforgetful functor.

Proof. It is a consequence of Proposition 7.1.2.2.

Notice that if we let (L, ∂′)|(K, ∂) be an extension of differential fields, and a dif-ferential module (V,∇) ∈ Ob(DiffModK), we can extend (V,∇) to an element ofDiffModL taking (V ⊗K L,∇L), where ∇L has the same connection matrix as ∇.We can now give the definition of a Picard-Vessiot extension.

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Definition 7.2.1.4. Let (V,∇) be a differential K-module. A Picard-Vessiot exten-sion for (V,∇) is an extension of differential fields (L, ∂′)|(K, ∂) such that

1. The constant field of both differential fields coincide, k(K,∂) = k(L,∂′).

2. V ⊗K L is generated as an L-vector space by its horizontal vectors.

3. The coordinates of the elements of (V ⊗KL)∇L in any L-basis of V ⊗KL comingfrom a K-basis of V generate the field extension L|K.

Notice that the second condition is stable under tensor products and subquotients.In fact, if it holds for a vector subspace W of V , there is an isomorphism

(W ⊗K L)∇ ⊗k L∼−−−→W ⊗K L,

where k is the constant field of L. Then, if L|K is a Picard-Vessiot extension of Kfor (V,∇), the functor ωL : ⟨(V,∇)⟩⊗ → fVeck that sends each V to (V ⊗K L)∇ is afibre functor. Also, thanks to the previous isomorphism, we have an algebraic grouphomomorhpism

Spm(L)→ Isom⊗K(ωL ⊗k K,ω|⟨(V,∇)⟩⊗).

This argument eventually leads us to the proof that if the constant field is algebraicallyclosed then there exists a Picard-Vessiot extension L for (V,∇). In order to do so,we must introduce another common notion in algebraic group theory.

Definition 7.2.1.5. Let G be a linear algebraic group over an algebraically closedfield of characteristic 0. A G-torsor Z over a field K ⊃ k is an affine algebraic varietyover K with a G-action, that is, a morphism GK ×K Z → Z that sends (g, z) to zgand such that for all x ∈ Z(K) and g, h ∈ G(K), z1 = z, z(gh) = (zg)h and themorphism GK ×K Z → Z ×K Z given by (g, z) 7→ (zg, z) is an isomorphism.

Remember that the field of rational functions of a torsor P is defined as the field offractions of Γ(P,O), and also that if P is irreducible, the field of rational functionsis integral.

Proposition 7.2.1.6. The isomorphism (W⊗KL)∇⊗kL∼−−−→W⊗KL of the previous

observation identifies L with the field of rational functions of Isom⊗K(ωL⊗kK , ω|⟨(V,∇)⟩⊗).

Proof. Let P be the GK-torsor Isom⊗K(ωL⊗kK , ω|⟨(V,∇)⟩⊗). The morphism

Spm(L)→ Isom⊗K(ωL⊗kK , ω|⟨(V,∇)⟩⊗)

can be translated in terms of k-algebras to

Γ(P,O)→ L.

The kernel I of the morphism is a ∇-stable ideal, because the kernel of a differ-ential morphism is ∇-stable. Thanks to 9.3.2 in [Del], I = 0 and so the previousmorphism is injective. The G-torsor P is hence contained in the GL(ωL(X))-torsorIsom⊗(ωL⊗kK , ω|⟨(V,∇)⟩⊗), and the fact that Γ(P,O) generates L gives us the secondpoint in 7.2.1.4.

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As a result, we obtain the following corollary.

Corollary 7.2.1.7. If the constant field k is algebraically closed, then there existsa Picard-Vessiot extension L|K for (V,∇). Furthermore, it is unique up to K-differential isomorphism.

Proof. Since a fiber functor ω0 : ⟨(V,∇)⟩⊗ → fVeck exists thanks to 5.3.0.12, thefield of rational functions of the previously introduced torsor P , that we denote asL, is a Picard-Vessiot extension of K, because for (W,∇) ∈ Ob(⟨(V,∇)⟩⊗) and a ∈ω0(X), since p(a) is a horizontal section of W on P , we have an isomorphism betweenω0(W )

∼−→ (W ⊗K L)∇ via a 7→ p(a), and also ω0(W ) ⊗k L∼−→ W ⊗K L. In the case

when W = V , point 1. in 7.2.1.4 is clear, point 2. of the same definition followsfrom the inclusion P → Isom⊗(ω0(X), X ⊗ K) and point 3. follows from point (ii)in 9.3 of [Del]. Thanks to the first isomorphism given by a 7→ p(a), ω0 is canonicallyisomorphic to ωL, and we have proven the result.

7.2.2 The differential Galois algebraic groupFirst, let us generalize Proposition 7.1.2.1, so that we can apply everything we knowto the category DiffModK .

Corollary 7.2.2.1. The category DiffModK of differential modules over K is anabelian, rigid symmetric monoidal k-linear category.

Proof. All we have to do is define the same tensor product, that is, for every (V,∇)and (V ′,∇′) in Ob(DiffModK), we define the tensor product of (V,∇) and (V ′,∇′)as (V ⊗ V ′,∇⊗∇′), where ∇⊗∇′ is given by the same rule in 7.1.2.1,

(∇⊗∇′)(s⊗ s′) := ∇(s)⊗ s′ + s⊗∇′(s′), ∀s⊗ s′ ∈ V ⊗ V ′.

The notion of the dual of (V,∇), ∇∗ on V ∗ = HomModK(V,K), is given by the rule

∇∗(ϕ)(v) = ∂ϕ(v)−ϕ(∇(v)). Following the same reasoning, we obtain the result.

Notice that if we fix a base differential field (K, ∂) and L a field that satisfies point 2.in Definition 7.2.1.4, each object of the full subcategory ⟨(V,∇)⟩⊗ satisfies the samecondition, thanks to the abelian and tensor structure on DiffModK . Hence, each Las the latter defines a fibre functor ωL on ⟨(V,∇)⟩⊗ given by (W,∇) 7→ (W ⊗K L)∇L .In fact, thanks to the existence and uniquenes (up to isomorphism) of Picard-Vessiotextensions for differential modules and the isomorphism G

∼−→ Aut⊗(ω) show thatPicard-Vessiot extensions for a differential module (V,∇) correspond bijectively toneutral fibre functors on ⟨(V,∇)⟩⊗.Furthermore, thanks to 5.3.0.12, we see that over an algebraically closed field k, onan abelian, rigid symmetric monoidal k-linear category tensor generated by a singleelement, a neutral fiber functor always exists into fVecK for an extension K|k. Noticethat this shows that Picard-Vessiot extensions always exist for differential modulesover differential fields with algebraically closed constant field.

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Definition 7.2.2.2. Let (L, ∂) be a Picard-Vessiot extension for a differential module(V,∇). We define the differential Galois algebraic group as

Gal(V,∇) := Aut⊗(ωL),

where ωL is the fiber functor of ⟨(V,∇)⟩⊗ given by (W,∇) 7→ (W ⊗K L)∇L .

Thanks to theorem 6.2.0.7, we have that Gal(V,∇) is an affine algebraic k-group.The reader who is familiarized with differential Galois theory should notice at thispoint that this is not exactly the same definition that can be found in any classicalapproach, which is the following.

Definition 7.2.2.3. If L|K is a differential extension of fields, the group of differ-ential K-automorphisms of L is called differential Galois group and we denote it asGal∂(L|K).

Our goal now is to prove that we can endow the latter group with the structure ofan affine algebraic group and also that it is isomorphic to Gal(V,∇) for a certaindifferential module (V,∇). So let us start by solving the first problem we mentioned.Let A ∈ Ob(Algk). We can extend the derivation ∂ of L by setting

∂′ : L⊗k A −→ L⊗k Af ⊗ r 7−→ ∂f ⊗ r

So we can define a group functor by setting Gal∂(L|K) := Aut∂WK(WL), where

Aut∂K⊗kA(L⊗kA) is the group of K⊗kA-algebra automorphisms of K⊗kA commuting

with ∂. When the differential field extension L|K is known, we will simply denoteit as Gal∂. On the other hand, for any f ∈ HomAlgk

(A,B), set Gal∂(L|K)(f) ∈HomAlgk

(Gal∂(A),Gal∂(B)) as the map that sends each automorphism of L⊗k A tothe automorphism of L⊗k B induced by base change via f . Hence, we have defineda group functor Gal∂ : Algk → Grp.

Proposition 7.2.2.4. Let L|K be a Picard-Vessiot extension for a differential module(V,∇). The previously defined functor Gal∂(L|K) defines an affine algebraic k-group.

Proof. In order to show this, we must find R ∈ Ob(Algk) such that Gal∂(L|K) =Spm(R). We see that A is in fact a quotient of Γ(GL[L:K],O). Let n := [L : K], andfix a K-basis of V just like in point 3. of 7.2.1.4, so that we can write the coordinatesof a k-basis of horizontal vectors in a matrix (fij) ∈ GLn(k). Notice that for anyσ ∈ Gal∂(k), σ multiplies (fij) by a matrix Mσ ∈ GLn(k). Since fij generate L overK, this determines σ and, analogously, for any σA ∈ Gal∂(A), where A ∈ Ob(Algk),we can find a matrix MσA ∈ GLn(A) giving a correspondence between elements ofGal∂(A) and GLn(A). On the other hand, there is a K-algebra exhaustive morphism

ϕ : B ⊗k K → L

induced by the map that sends xij ∈ Γ(GLn,O) to fij and that is also compatiblewith the derivations of B and L. Its kernel, P := Ker(ϕ) ⊆ B ⊗k K is a maximalideal and each L ⊗k A-automorphism σA ∈ Gal∂(A) canbe lifted to a morphism

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σA : B ⊗k K ⊗k A→ L⊗k A such that σA(PA) = 0, where PA is the ideal generatedby P in B ⊗k K ⊗k A. Now, let eλλ∈Λ be a k-basis of L. We can write

σA(b⊗ r ⊗ a) =∑λ∈Λ

ασAλ (b⊗ r ⊗ a)eλ, ∀b⊗ r ⊗ a ∈ B ⊗k K ⊗k A.

The sum is finite and each ασAλ (b⊗r⊗a) ∈ A, for all b⊗r⊗a ∈ B⊗kK⊗kA. Hence,the condition we must impose is that

ασAλ (b⊗ r ⊗ a) = 0, ∀b⊗ r ⊗ a ∈ PA.

It suffices to show it for a finite system of generators of P , p1, . . . , pm. Let A = B,and let σB be the automorphism corresponding to MσB . Also, let

R :=B

⟨ασBλ (pl) : 1 ≤ l ≤ m,λ ∈ Λ⟩.

For a given A ∈ Ob(AlgK) and σA corresponding to MσA , let B → A be the morphismthat sends xij to (MσR)ij. Since σA comes from a σA satisfying σA(PA) = 0, B → Afactors through R and therefore we have that Gal∂ = Spm(R).

Now, in order to achieve our second goal, that is, seeing that Gal(V,∇) is isomorphicto Gal∂(L|K), we must construct a functor between ⟨(V,∇)⟩⊗ and Rep(Gal∂(L|K)).In order to do so, let (V,∇) ∈ Ob(DiffModK) and let L|K be a Picard-Vessiotextension for (V,∇). We can extend any σ ∈ Gal∂(L|K)(k), i.e, any K-automorphismof L σ : L → L as σ : V ⊗K L → V ⊗K L and it will satisfy σ

((V ⊗K L)∇L

)=

(V ⊗KL)∇L , which shows that ωL : DiffModK → fVeck can be regarded as a functoron DiffModK into the category of finite dimensional representations of Gal∂(L|K)(k).On the other hand, for each (W,∇) ∈ Ob(⟨(V,∇)⟩⊗), we can define a k-algebra functorA 7→ (W ⊗K L⊗K A)∇L⊗KR , gives us a functor

Ω : ⟨(V,∇)⟩⊗ → Rep(Gal∂(L|K))

In order to prove the next proposition, we must see a previous result.

Lemma 7.2.2.5. Let (K, ∂) be a differential field with constant field k = k and L|Ka Picard-Vessiot extension for (V,∇). Then, the differential field of Gal∂(L|L)(k)-invariant elements of L is K.

Proof. Firstly, thanks to the proof of Proposition 7.2.2.4, we know that L is the fieldof fractions of R := Γ(GL[L:K],O)/P , where P is a maximal differential ideal. Hence,let b

c∈ L \ k with b, c ∈ R and d := b ⊗ c − c ⊗ b ∈ R ⊗k R. Thanks to point A.15

in [P], we have d = 0, and by A.16 from the same book, R ⊗k R has no nilpotentelements because Char(k) = 0. Let J be a maximal diferential ideal in (R⊗kR)[1/d].Also, set a derivation on the latter differential ring via

D(r ⊗ r′) := ∂r ⊗ r′ + r ⊗ ∂r′.

Let ϕi : R→ N := (R⊗kR)[1/d]J

given by the tensor product. The images of each ϕi aregenerated by fundamental matrices of the same matrix associated to the differential

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equation that the latter extension defines. Therefore, both images are equal to acertain S ⊂ N and ϕi : R → S are in fact isomorphism. This shows the existence ofan element σ ∈ Gal∂(L|K)(k) satisfying ϕ1 = ϕ2 σ. The image of d in N is equalto ϕ1(b)ϕ2(c) − ϕ1(c)ϕ2(b), and since the image of d in N is non-zero by previousobservations, we have ϕ1(b)ϕ2(c) = ϕ1(c)ϕ2(b). Hence, ϕ2((σ(c))b) = ϕ2((σ(b))c) so(σ(b))c = (σ(c))b, which shows that σ(b/c) = b/c.

So now we have all the tools we need in order to show the following proposition, whichgives us the relationship we were looking for.

Proposition 7.2.2.6. Let (V,∇) be a differential module and let L|K be a Picard-Vessiot extension for (V,∇). If the constant field k is algebraically closed, then thefunctor Ω : ⟨V,∇⟩⊗ → Rep(Gal∂(L|K)) defines an equivalence of neutral Tannakiancategories.

Proof. Our goal is to show that Ω is fully faithful. Let (W,∇), (W ′,∇) ∈ Ob(⟨V,∇⟩⊗).Thanks to the rigidity of ⟨V,∇⟩⊗, we have an isomorphism

HomDiffModK((W,∇), (W ′,∇)) ≃ HomDiffModK

(K, (W∨ ⊗K W ′,∇)).

Analogously, for the corresponding representations of Gal∂ we may assume that(W,∇) is the field K endowed with the trivial connection. We know that the el-ements of HomDiffModK

(K, (W ′,∇)) correspond bijectively to W ′∇. Since k = k, theelements of ωL(HomDiffModK

(K, (W ′,∇))) correspond to elements of (W ′ ⊗K L)∇L

invariant by Gal∂(L|K)(k). Using the previous lemma, the Gal∂(L|K)(k)-invariantelements in W ′ ⊗K L are exactly the elements of W ′, hence we obtain the horizontalelements of W ′. Finally, in order to see that it is exhaustive, taking into accountpoint 3 of 7.2.1.4, ωL maps (V,∇) to a faithful representation of Gal∂(L|K), or inother words, Gal∂(L|K) → GLn(k) is injective. Using Lemma 6.5.16 of [Sza], theresult follows.

So we conclude this chapter by proving the following classical theorem.

Theorem 7.2.2.7. Let (K, ∂) be a differential field with constant field k = k and letL|K be a Picard-Vessiot extension for the differential module (V,∇) over K. Themap

H(k) 7→ LH(k)

induces a bijection between closed subgroups of Gal∂(L|K)(k) and differential subfieldsof L containing K. Closed normal subgroups correspond to Picard-Vessiot extensionsof K and the associated differential Galois group is (Gal∂(L|K)/H)(k).

Proof. Let M be an intermediate differential field between K and L. Then, L isa Picard-Vessiot extension for the differential module obtained by the change ofbasis, (VM ,∇M). This identifies Gal∂(L|M)(k) with a closed subgroup H(k) ofGal∂(L|K)(k), just like it is closed in GLn(k) with the same GLn as for Gal∂(L|K)(k).Using Lemma 7.2.2.5, we have LH(k) = M . Analogously, a closed subgroup H(k) ⊂Gal∂(L|K)(k) fixes some M with Gal(VM ,∇M)(k) ≃ H(k). On the other hand, givena closed normal algebraic subgroup H ⊂ Gal∂(L|K), letM∈ Rep(Gal∂(L|K)) with

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kernel H. The full subcateogry ⟨M⟩⊗ is equivalent to Rep(Gal∂(L|K)(k)) by Lemma6.5.16 of [Sza] and on the other hand to ⟨W,∇⟩⊗ for some (W,∇) ∈ Ob(⟨V,∇⟩⊗).Since L satisfies points 1. and 2. of 7.2.1.4 for (W,∇), and by 3. of the same def-inition. it contains a Picard-Vessiot extension M for (W,∇). By construction, wehave Gal(W,∇) ≃ Gal(V,∇)/H and Gal∂(L|M)(k) ≃ H(k) by the first part of theproof.

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