Affine Group Schemes
Boas Meijer
July 11, 2017
Bachelor Project Mathematics Supervisor: Dr. Lance Gurney
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Korteweg-de Vries Instituut voor Wiskunde
Abstract
This thesis gives an introduction to affine group schemes and in particular their cor-respondence with Hopf Algebras. In the first chapter, a categorical foundation is laid and group objects are discussed. Besides this category theory, algebras over a field are introduced and those give rise to a mathematical structure called the affine schemes. These are co-representable functors from the category of algebras over a field and group objects on these affine schemes are called affine group schemes. In the third chapter, affine group schemes are discussed through multiple examples and their correspondence to Hopf algebras is explained. In the final chapter, this correspondence is built upon by introducing a way to classify Hopf algebras named Cartier duality.
Title: Affine Group Schemes Authors: Boas Meijer,
Supervisor: Dr. Lance Gurney
Second grader: Dr. Hessel Bauke Posthuma Date: July 11, 2017
Korteweg-de Vries Instituut voor Wiskunde Universiteit van Amsterdam
Science Park 904, 1098 XH Amsterdam http://www.science.uva.nl/math
Contents
1 Introduction 4
2 Basic Category theory 5
2.1 Categories . . . 5 2.2 Functors . . . 8 2.3 Yoneda’s Lemma . . . 10 2.4 Group objects . . . 11 3 Affine Schemes 13 3.1 Algebras . . . 13 3.2 Affine schemes . . . 15
4 Affine Group Schemes 18 4.1 Affine group schemes . . . 18
4.2 Hopf algebras . . . 20
4.3 Correspondence of Hopf algebras and affine group schemes . . . 24
5 Cartier Duality 27 5.1 Dual vector spaces . . . 27
5.2 Cartier Duality . . . 29
6 Conclusion 34
7 Popular summary 35
1 Introduction
In elementary algebra, groups are one of the first and simplest algebraic structures discussed. Each group has an underlying set and if this set has an extra structure, such that this structure is compatible with the group structure, this gives a new notion of a group. A well known example of such a group is a topological group A. In this case, every underlying set is a topological space and the group operation is a continuous map from the product spaceA× A to A.
This is a general phenomenon which can be used to define group objects in any cat-egory. An example of such group objects, the group objects in the category of affine schemes, is the main subject of this thesis.
To understand what it means for an object to have a group structure, some basic knowledge of category theory is needed. The bare minimum about categories and func-tors is discussed such that an understanding of what it means for an object to have a group structure can be obtained.
In the extent of this thesis, the focus will be on the group objects in the category of affine schemes. Therefore, it needs to be stated what affine schemes are and this is done in the second chapter. There will be a short section on algebras over a field and what categorical properties they hold, followed by the definition and some examples of what affine schemes are.
The first two chapters lead to the definition of an affine group scheme. This is the main topic of the bachelor project and in the last two chapters, some basic properties and ways of constructing affine group schemes are covered. First some examples are given to get a feel of what affine group schemes are. Second an algebraic structure is introduced called the Hopf algebras over a field. Hopf algebras form a category which is equivalent to the opposite category of affine group schemes. Due to this correspondence, they can be used to obtain affine group schemes that are difficult to discover through the definition of a group scheme alone.
The last chapter will discuss a key property used to classify finite affine group schemes called Cartier duality. The general notion of duality will be discussed as well as the Cartier dual of the finite affine group schemes Z/nZ and αp.
2 Basic Category theory
To understand what it means for some object to have a group structure, it is necessary to structurize and categorize these objects and their respected properties and this is done through categories. Though category theory is a mathematical field of its own, it is especially useful in generalizing and connecting mathematical structures. In the case of this thesis, it will be used to obtain a definition of what it means for a mathemati-cal structure to also have a group structure, as well as to compare seemingly different structures with each other. In this chapter, the notes from Lenny Taelman (Taelman, 2017) and Joost van Oosten (van Oosten, 2002) were used as well as the book Category theory by Steve Awodey (Awodey, 2010).
2.1 Categories
Categories are the main way to distinguish and “group” different objects. Throughout this section, they will be properly defined and familiarized.
2.1.1 Definition & Examples
2.1.2 Definition. A category C consists of: • Objects: a class obC
• Morphisms: for every two objects X and Y in C, a class HomC(X, Y )
• Identity: for every object X in C, a morphism idX ∈ HomC(X, X)
• Composition: for every X, Y and Z in obC, f ∈ HomC(X, Y ) and g∈ HomC(Y, Z),
there is a composition
HomC(X, Y )× HomC(Y, Z)→ HomC(X, Z), (f, g)7→ gf,
subject to the following conditions:
– Associativity: forW, X, Y, Z in obC and for f ∈ HomC(W, X), g∈ HomC(X, Y )
and h∈ HomC(Y, Z), h(gf ) = (hg)f
– Unit: for X and Y in obC and for f ∈ HomC(X, Y ), f idX =f and idYf = f
In this definition, it is easiest to think of a class as a kind of set. However, the term class is used for set theoretical purposes as there is no such thing as the set of all sets but there does exist a category with a class of all sets. Objects can be a wide range of things like sets, groups, spaces or morphisms. Often, a class of objects of one category
can even be thought of as an object in another category’s class of objects. As you might have already noticed, there are many familiar mathematical examples of categories. 2.1.3 Example. The most basic category is the category of sets (Set). For this category, the class of objects is the class of all sets and the morphisms are all maps between two sets. Each set has an obvious identity map and the usual compositiongf := g◦ f holds. 2.1.4 Example. Groups form a category denoted Grp. In this category, the class of objects consists of all groups and for A and B in obGrp, HomGrp(A, B) consists of
all group morphisms from A to B. As associativity and the existence of an identity morphism hold for group morphisms, groups with group morphisms form a category. 2.1.5 Example. To illustrate the broad notion of categories, it is interesting to see that a group itself can also be thought of as a category.
If G is an arbitrary group with operation + and identity element 0, then there exists a one object categoryN with obN = {∗} and HomN(∗, ∗) = G with + as composition
and id∗ = 0. As each group is associative and has an identity element, composition is
associative andid∗ is an identity morphism making this a category.
Categories can be found anywhere in higher mathematics. As shown above, groups, one of the basic concepts in algebra, form a category. In the diagram below, some other well known mathematical structures that can be described as categories are stated.
Category name Objects Morphisms Set Sets Maps between sets Ring Rings Ring morphisms
RMod(R) Left (right) R-modules R-module morphisms
Ab Abelian groups Group morphisms CRing Commutative Rings Ring morphisms Top Topological spaces Continuous maps Grph Graphs Graph morphisms Pos Partially ordered sets Monotone functions Veck Vector spaces over field k Linear transformations
2.1.6 Definition. The opposite category of a category C is the category Copp with:
• obCopp:= obC
• For X, Y in C, HomCopp(X, Y ) := HomC(Y, X)
Where forX, Y, Z in obCopp,f ∈ Hom
Copp(X, Y ) and g ∈ HomCopp(Y, Z), composition is
given by
HomCopp(X, Y )× HomCopp(Y, Z)→ HomCopp(X, Z), (f, g)→ fg.
Now that we have defined the notion of a category, we can try to extend certain concepts we know about sets to categories.
2.1.7 Initial and Final objects
In the category of sets, there are two unique sets with interesting properties. These are the one element set{∗} and the empty set ∅ and they can be characterized by the class of all maps from and to them. To clarify, the only set that has a unique map to every other set is the empty set, and the only set that has a unique map from every set to that set is the one element set. This concept can be generalized to categories as follows. 2.1.8 Definition. An object X in a categoryC is called an initial object if for all Y ∈ obC, there is a unique morphism from X to Y .
2.1.9 Definition. Similarly, an objectY in a category C is called a final object if for all X∈ obC, there is a unique morphism from X to Y .
Listed below are some categories with their initial and final objects. As you can see, they do not always exist.
Category Initial Object Final object Grp {e} {e}
Ring Z {0} Field - -ModR {0} {0}
Top ∅ {∗}
2.1.10 Remark. Initial and final objects are unique up to unique isomorphism and X is the initial object inC if and only if X is the final object in Copp.
2.1.11 Products
For every two sets, we can construct a new set by taking the Cartesian product. This is simply the set consisting of all ordered pairs of the two sets. Like initial and final objects, this is a concept that can be generalized to categories.
2.1.12 Definition. ForX, Y objects in a category C, a product of X and Y consists of: • An object P in C
• Morphisms πX :P → X and πY :P → Y
Such that for all objectsT ∈ C and all morphisms fX ∈ Hom(T, X) and fY ∈ Hom(T, Y ),
there exists a unique morphismh : T → P for which the diagram below commutes. X T fX 77 fY '' h //P πX OO πY Y
2.1.13 Remark. If Q is an object and ιX, ιY are morphisms in a category C, then
(Q, ιX, ιY) is the coproduct of X and Y in C if and only if (Q, ιX, ιY) is the product
of X and Y inCopp.
Listed below are examples of products and coproducts for some categories.
Category Product Coproduct
Set Cartesian product Disjoint unionXq Y with i1,i2 the inclusion maps
Top Cartesian product Disjoint unionXq Y with the natural topology Ab Direct product Direct sum
ModR Direct product Direct sum
2.2 Functors
What hopefully became clear in the previous section is that the definition of a category is very broad. Categories can take on many different forms and are found whenever you speak of a mathematical structure. This raises the question in what way categories can be compared and the main tool used to do this is a type of mapping called a functor.
2.2.1 Definition & Examples
2.2.2 Definition. For categories C and D, a functor F : C → D consists of: • for every object X ∈ C, an object F (X) ∈ D
• for every morphism f ∈ HomC(X, Y ), a morphism F (f ) ∈ HomD(F (X), F (Y ))
Such that:
– for X∈ obC, F (idX) =idF (X)
– for X, Y, Z ∈ obC, f ∈ HomC(X, Y ) and g∈ HomC(Y, Z), F (gf ) = F (g)F (f )
2.2.3 Example. Functors can be thought of as morphisms between categories. In fact, a new category Cat can now be described with:
• obCat is the class of all categories for which the classes are sets
• For every C, D ∈ obCat, HomCat(C, D) is the class of all functors from C to D
This is a category as for categories B, C and D, and for functors F : B → C and G :C → D, there is a composite functor
GF :B → D with (GF )(X) = G(F (X)) and (GF )(f) = G(F (f)).
2.2.4 Example. Many categories have as objects all sets with some algebraic structure and as morphisms all maps that preserve this structure. For such categories, there is a functor that maps to the category of sets and simply “forgets” the algebraic structure. Such a functor is called a forgetful functor.
For example, the functor F : Ring → Set, which sends a ring X to the set X and a ring morphism f to the morphism of sets f is a forgetful functor. Other well known categories satisfying the above classification are ModR, Top and Grp.
2.2.5 Example. For an object X in a category C, a functor F : C → Set can be con-structed such that:
• For Y ∈ obC, F (Y ) := HomC(X, Y )
• For Z1, Z2 ∈ obC and f ∈ HomC(Z1, Z2), F (f ) : HomC(X, Z1) → HomC(X, Z2),
g→ fg
This functor is called the Hom functor and is denoted hX := Hom
C(X,−). There is
also a contravariant version of this functor denotedhX := HomC(−, X) : Copp→ Set.
2.2.6 Natural transformations
As we have seen before, objects can be described and compared to each other through morphisms and categories through functors. The natural way to compare functors is through natural transformations (sometimes referred to as functor morphisms).
2.2.7 Definition. LetC and D be categories and let F : C → D and G : C → D be functors. A natural transformationη : F → G consists of a morphism ηX :F (X)→ G(X) for
every object X ∈ C, such that for all Y ∈ obC and all f ∈ HomC(X, Y ), the following
diagram commutes. F (X) ηX F (f ) //F (Y ) ηY G(X) G(f ) //G(Y )
2.2.8 Definition. A natural transformation η : F → G with F and G functors from categories C to D is called a natural isomorphism if for every X in C, ηX :F (X) →
G(X) is an isomorphism.
2.2.9 Example. Consider the class of all functors from a category C to a category D. This class forms a categoryDC where forF and G in obDC, the natural transformations
from F to G are the morphisms from F to G, Nat(F, G) = HomDC(F, G). So for any
two categories, we can construct a new category consisting of functors from one to the other.
2.2.10 Equivalence of Categories
As mentioned earlier, functors play an important role in distinguishing categories. As categories are defined to be a class of objects with a class of morphisms subject to some properties, there are categories that are categorically identical but that are technically not the same category. This equivalence is described below.
2.2.11 Definition. For categories C and D, Let F : C → D be a functor. If for every X, Y ∈ C the map
HomC(X, Y )→ HomD(F X, F Y ), f 7→ F f
is surjective, we say that the functor F is full. If the maps are injective, we say F is faithful. Furthermore, the functor F : C → D is essentially surjective if for every Z ∈ obD, there is an object X ∈ C such that F (X) ∼=Z.
2.2.12 Definition. A functor F :C → D is an equivalence of categories if there is a functor G :D → C with natural isomorphisms : F G → idD and η : GF → idC.
2.2.13 Remark. A functor is an equivalence of categories if and only if it is full, faithful and essentially surjective.
2.3 Yoneda’s Lemma
In this section, an important lemma regarding the characterization of objects in a cat-egory is discussed through the Hom functor described in example 2.2.5. This lemma is called Yoneda’s lemma and it is similar to Cayley’s theorem. Where Cayley’s theorem states that every group is a subgroup of a permutation group, Yoneda’s lemma says that every category is a full subcategory of a functor category.
2.3.1 Definition. A functorF :Copp→ Set is called representable if there is an object
X∈ C such that F is naturally isomorphic to HomC(−, X) = hX.
2.3.2 Definition. Similarly, a functor G :C → Set is called co-representable if there is an object X∈ C such that G is naturally isomorphic to HomC(X,−) = hX.
2.3.3 Example. The forgetful functor,F : Grp→ Set, is co-represented by Z.
For any groupG and any f ∈ HomGrp(Z, G), f is completely determined by where it
sends 1. Therefore, HomGrp(Z, G) ∼=G ∼=F (G) as sets and F ∼=hZ.
2.3.4 Definition. For a categoryC, the Yoneda functor is the functor h : C → SetCopp, such that forA and B ∈ obC and for f ∈ HomC(A, B),
A7→ hAand f 7→ hf.
Here hf denotes the natural transformation from hA tohB.
2.3.5 Lemma. (Yoneda’s lemma) The Yoneda functor is fully faithful and injective on objects such that for objects A and B in a category C, there is a natural isomorphism
HomSetCopp(hA, hB) ∼=hB(A) = HomC(A, B).
There is also a co-Yoneda functor and a co-Yoneda lemma described similarly. 2.3.6 Remark. An application of the Yoneda lemma is to show that two objects in a category are isomorphic. To specify, if A, B ∈ obC for some category C, then to show thatA ∼=B, it suffices to show that hA∼=hB. As hom functors are sometimes easier to
2.4 Group objects
To begin this section, a new definition of a familiar algebraic structure is given. This is done in order to extend the definition from the category of sets to arbitrary categories. 2.4.1 Definition. A group is a setG with maps m : G×G → G (operation), e : {∗} → G (unit) and inv : G → G (inverse), where {∗} is the one element set, such that the following diagrams commute.
G× G × G (m,idG) (idG,m) //G× G m (Associative) G× G m //G G (e◦q,idG) idG (( (idG,e◦q) //G× G m (U nit) G× G m //G G (inv,idG) q //{∗} e G q oo
(idG,inv) (Inverse)
G× G m //Goo m G× G
In the above diagrams, q is the unique map from the set G to a one element set {∗}. So a group is an object together with three morphisms in the category of sets such that the above diagrams commute. Using the obtained categorical knowledge, it is now possible to define what it means for an object in a category to have a group structure. 2.4.2 Definition. A group object (X, m, inv, e) in a categoryC with final object ∗ and finite products, is an object X with morphisms m : X × X → X, e : ∗ → X and inv : X → X such that the following diagrams commute.
X× X × X (m,idX) (idX,m) //X× X m (Associative) X× X m //X X (e◦q,idX) idX (( (idX,e◦q) //X× X m (U nit) X× X m //X X (inv,idX) q //{∗} u X q oo
(idX,inv) (Inverse)
X× X m //Xoo m X× X
Here, q : X → {∗} is the unique map from an object X to the final object {∗}, m represents the group operation,inv sends an element in X to its inverse element and e sends {∗} to the identity element in X.
2.4.3 Remark. As one might expect, in the category of sets, a group object is simply a group and in the category of topological spaces, a group object is a topological group. 2.4.4 Example. In the category of groups, the abelian groups are the group objects.
2.4.5 Proposition. If (G, +) is a group and (G, m, e, inv) is a group object in the category of groups, then the identities and operations coincide.
Proof. Let 1G be the identity element of (G,·) and e(∗) the identity element of the
group object (G, m, e, inv). As m : G× G → G is a group morphism, for a, b, c, d ∈ G, m(a· c, b · d) = m(a, b) · m(c, d) such that
e({∗}) = m(e({∗}), e({∗})) = m(e({∗}) · 1G, 1G· e({∗}))
=m(e({∗}), 1G)· m(1G, e({∗})) = 1G· 1G= 1G
and
m(a, b) = m(a· 1G, 1G· b) = m(a, 1G)· m(1G, b) = a· b.
It follows (G, m, e, inv) and (G,·) represent the same group.
Now say (A,·) ∈ obGrp and (A, m, e, inv) is a group object. In the previous proposi-tion, it was shown that the group structures coincide such that fora∈ A, inv(a) = a−1. Asinv is also a group morphism,
y−1· x−1 = (x· y)−1=inv(x· y) = inv(x) · inv(y) = x−1· y−1 such that (A,·) is an abelian group.
2.4.6 Example. If C is a category with terminal object ∗ and finite products, then for A and B in obC, hA× hB ∼= hA×B as the functor hA preserves products. Now due to
Yoneda’s lemma, (A, m, inv, u) is a group object in C if and only if (hA, hm, hinv, hu) is
a group object in SetCopp.
To generalize the concept of a group, it was essential to first introduce categories as they are the main way to group and differentiate mathematical structures. Functors are the mappings used to compare categories and in the extend of this thesis, these functors, as well as Yoneda’s lemma and group objects are used regularly.
3 Affine Schemes
To explore affine group schemes, it is first necessary to get to know general affine schemes. In this chapter, affine schemes will be defined and some examples will be given but before this is done, a new algebraic structure needs to be defined. The lecture notes from Ken Brown’s course on Hopf algebras (Brown, 2007) and the syllabus from James Milne on affine group schemes (Milne, 2012) are the main sources for this chapter.
3.1 Algebras
Algebras over a field are rings with scalar multiplication by the given field. In this section, they will be properly defined and their similarities will be discussed.
3.1.1 Definition. The tensor product of vector spaces M and N over a field k is a vector spaceT with a k-bilinear map g : M× N → T such that for a vector space A and k-bilinear map f : M × N → A, there is a unique k-linear map h : T → A with f = hg.
M× N g // f T h {{ A
The pair (T, g) is unique up to unique isomorphism and we denote it by M ⊗kN := T
and for (x, y)∈ M × N, x ⊗ y := g(x, y).
3.1.2 Remark. If M1, M2, N1, N2 are vector spaces over field k and f : M1 → M2 and
g : N1→ N2 are k-linear maps, then there is a corresponding bilinear map M1× N1 →
M2 ⊗k N2, (m1, n1) 7→ f(m1) ⊗ g(n1). Moreover, this map induces a k-linear map
f⊗ g : M1⊗kN1→ M2⊗kN2,m1⊗ n1 7→ m2⊗ n2.
3.1.3 Definition. A k−algebra is a vector spaceA over a field k with k-linear maps m : A⊗ A → A and e : k → A,
such that the diagrams below commute.
A⊗ A × A (m,idA) (idA,m) //A⊗ A m (Associative) A⊗ A m //A A⊗ A m (( A⊗ k (id⊗e) oo ∼ (U nit) k⊗ A (e⊗id) OO ∼ //A
3.1.4 Remark. An intuitive restatement of the above definition is that an algebra over a fieldk is simply a ring A with a ring morphism k→ A.
3.1.5 Definition. A k-algebra (A, m, e) is commutative if the following diagram com-mutes. A× A m swap //A× A m (Commutative) A id //A
Here the the morphismswap : A× A → A × A swaps the two factors.
3.1.6 Definition. If (A, mA, eA) and (B, mB, eB) are k-algebras, then a k-linear map
f : A→ B is a k-algebra morphism if the following diagram commutes.
A⊗ A mA f ⊗f //B⊗ B mB A f //B
3.1.7 Example. For an arbitrary group (G,•), a k-algebra kG called the group algebra can be constructed. kG is the set of all finite linear combinations of elements in G with coefficients ink. Addition and scalar multiplication on this set are defined such that for x =P g∈Gαgg, y =Pg∈Gβgg∈ kG and r ∈ k, (X g∈G αgg) + ( X g∈G βgg) = X g∈G (αg+βg)g and r X g∈G αgg = X g∈G (rαg)g.
It is easy to check that these operations makekG into a vector space over k. Now define thek-linear maps m : kG× kG → kG and e : k → kG such that
m((X g∈G αgg), ( X g∈G βgg)) = X g∈G,h∈G (αgβh)(g• h) and e(r) = r.
AsG and k are associative with identities, m is associative and e is a unit.
3.1.8 Example. The ring of polynomials over the fieldk (denoted k[X]) is a k-algebra with the ring morphismψ : k → k[X], r 7→ r for all r ∈ k representing scalar multiplication. 3.1.9 Proposition. If B is a k-algebra, then B ∼=k[Xb :b∈ B]/I for some ideal I.
Proof. Define the k-algebra morphism ϕ : k[Xb : b ∈ B] → B, Xb 7→ b. This map is
surjective by construction, thus B ∼= k[Xb : b ∈ B]/ ker ϕ. But as ker ϕ is an ideal,
I := ker ϕ such that,
B ∼=k[Xb:b∈ B]/ ker ϕ = k[Xb:b∈ B]/I.
As the algebras over a fieldk form a category, we can discuss some of their categorical properties.
Initial objects In Algk, the initial object is the k-algebra k.
If B ∈ Algk, then B ∼=k[Xb :b∈ B]/I for some ideal I. And if f ∈ HomAlgk(k, B),
then ask is a field, f is completely determined by where it sends 1k. Asf is a morphism
of k-algebras, 1k will be send to 1B such that there is only one possibility forf .
Final objects The final object is the zero algebra{∗}.
This is the case as the zerok-algebra morphism is the only morphism that sends every element from an arbitraryk-algebra to {∗}.
Products The product of two k-algebras A and B ∈ Algk is the cartesian product A× B with the k-algebra morphisms πA : A× B → A, (a, b) 7→ a and πB : A× B →
B, (a, b)7→ b.
The cartesian product of two k-algebras is again a k-algebra and for every k-algebra T , if f ∈ HomAlgk(T, A) and g∈ HomAlgk(T, B), then for t∈ T , the morphism h : T →
A× B, t 7→ (f(t), g(t)) satisfies that πA◦ h = f and πB◦ h = g.
Coproduct The coproduct of objectsA, B ∈ Algk is the tensor productA⊗kB with
thek-algebra morphisms ιA:A→ A⊗kB, a7→ a⊗1BandιB:B → A⊗kB, b7→ 1A⊗b.
A tensor product of twok-algebras is again a k-algebra as k-linear maps mA, eA, mB, eB
makingk-vector spaces A and B into k-algebras induce unique k-linear maps mA⊗ mB
and eA⊗ eB makingA⊗kB into a k-algebra.
Now to show that this tensor product is indeed the coproduct, consider a k-algebra T and morphismsf ∈ HomAlgk(A, T ) and g∈ HomAlgk(B, T ). Now define a bilinear map
H : A× B → T such that for all a ∈ A and b ∈ B, H(a, b) = f(a)g(b). H induces a uniquek-algebra morphism h : A⊗ B → C such that h(a ⊗ b) = f(a)g(b) and for which
h◦ ιA(a) = h(a⊗ 1) = mC(f (a), g(1)) = mC(f (a), 1) = f (a) and
h◦ ιB(b) = h(1⊗ b) = mC(f (1), g(b)) = mC(1, g(b) = g(b).
This makes (A⊗ B, ιA, ιB) the coproduct ofk-algebras A and B.
3.2 Affine schemes
Having defined algebras over a field, we can combine the previously obtained knowledge to discuss affine schemes. Just like with algebras, it happens that all affine schemes are of a similar form.
3.2.1 Definition. An affine k−scheme is a co-representable functor from the category of algebras over a fieldk to the category of sets.
3.2.2 Remark. Affine k-schemes form a category Affk with as objects the affine
k-schemes and as morphisms the natural transformations. As affine k-schemes are co-representable, by the co-Yoneda lemma, h : Algoppk → SetAlgk is fully faithful and
induces an equivalence of categories Algoppk ∼= Affk.
3.2.3 Proposition. Iff1, f2, . . . , fm∈ k[X1, . . . , Xn] for somen, m∈ Z, then the
func-torF : Algk→ Set with for every A and B in Algk and g∈ HomAlgk(A, B):
• F (A) = {x ∈ An|f
1(x) = f2(x) =· · · = fm(x) = 0}
• F (g) : F (A) → F (B), x = (x1, . . . , xn)7→ (g(x1), . . . , g(xn))
is an affinek-scheme.
Proof. First, it will be shown thatF is indeed a functor. The affine k-scheme property will be shown through the construction of two natural transformations between F : Algk → Set and hC : Algk → Set with C = k[X1, . . . , Xn]/(f1, . . . , fm) that are each
others inverse.
F : Algk→ Set is a functor as
F (idA) = [(x1, . . . , xn)7→ (idA(x1), . . . , idA(xn))]
= [(x1, . . . , xn)7→ (x1, . . . , xn)] =idF (A),
and for g∈ HomAlgk(A, B) and h∈ HomAlgk(B, C),
F (hg) = [(x1, . . . , xn)7→ (hg(x1), . . . , hg(xn))]
= [(y1, . . . , yn)7→ (h(y1), . . . , h(yn))]◦ [(x1, . . . , xn)7→ (g(x1), . . . , g(xn))]
=F (h)◦ F (g), wherey = (y1, . . . , yn)∈ F (B).
Now for allA∈ obAlgk we can define a morphism ϕA such that for allα∈ hC(A) =
HomAlgk(C, A),
ϕA:hC(A)→ F (A), α 7→ (α(X1), . . . , α(Xn)).
For every A and B∈ obAlgk and g∈ HomAlgk(A, B),
(F (g)◦ ϕA)(α) = F (g)(α(X1), . . . , α(Xn)) = (g◦ α(X1), . . . , g◦ α(XN))
=ϕB◦ (g ◦ α) = (ϕB◦ hC(g))(α).
This implies thatϕ : hC → F is a natural transformation.
Furthermore, as for everyA∈ obAlgkandx = (x1, . . . , xn)∈ F (A), there is a unique
morphismγx∈ hC(A) such that γ(Xi) =xi for 1≤ i ≤ n, we can define a morphism,
ψA:F (A)→ hC(A), x7→ γx.
Algk,x = (x1, . . . , xn)∈ F (A) and g ∈ HomAlgk(A, B),
(hC(g)◦ ψA)(x1, . . . , xn) =hC(g)(γx) =g◦ γx =γg(x)
=ψB(g(x1), . . . , g(xn)) = (ψB◦ F (g))(x1, . . . , xn).
To conclude this example, it needs to be shown thatψAis the two sided inverse ofϕA
for all A∈ Algk.
It holds thatϕA◦ ψA=idF (A) as forx = (x1, . . . , xn)∈ F (A),
(ϕA◦ ψA)(x) = ϕA(γx) = (γx(X1), . . . , γx(Xn)) = (x1, . . . , xn) =x.
Similarly, ψA◦ ϕA = idhC(A) as for α ∈ hC(A) and for 1≤ i ≤ n, γα(X)(Xi) = α(Xi))
such that
(ψA◦ ϕA)(α) = ψA(α(X1), . . . , α(Xn)) =γα(X) =α.
ThusF is co-represented by C = k[X1, . . . , Xn]/(f1, . . . , fm).
3.2.4 Example. The functor Ank : Algk → Set with for every A and B in Algk and
f ∈ HomAlgk(A, B):
• An
k(A) = An
• An
k(f ) : Ank(A)→ Ank(B), (a1, . . . , an)7→ (f(a1), . . . , f (an))
is an affinek-scheme called the affine n-space. This is indeed an affine k-scheme as by the previous proposition, Ank is co-represented by thek-algebra k[X1, . . . , Xn].
The affine 1-space is called the affine line and for an affinek-scheme T co-represented by some k-algebra B,
Hom(T, A1k) = Hom(hB, hk[X]) ∼= Hom(k[X], B) ∼=B (by Yoneda).
3.2.5 Example. Say B is a k-algebra. Then the functor
F : Algk→ Set, B 7→ {b ∈ B : b3 = 0}
is an affine k-scheme.
As {b ∈ B : b3 = 0} = {b ∈ B : f(b) = 0} with f(b) = b3, by proposition 3.2.3
we get that F (B) ∼= HomAlgk(k[X]/X
3, B). Therefore, F is isomorphic to the functor
HomAlgk(k[X]/X
3,−) such that F is an affine k-scheme.
In this chapter, a well known algebraic structure called the algebras over a field was introduced. This was done in order to define affine schemes, a topic we are not to interested in by itself. However, the category of affine schemes hold some interesting objects as will be shown in the next chapter.
4 Affine Group Schemes
In this chapter, the knowledge of the previous chapters is combined to define affine group schemes. They correspond to Hopf algebras, an algebraic structure that will also be introduced, and this correspondence will be broadly discussed throughout this chapter. Like in the previous chapter, the theory in this chapter is mainly based on the notes from James Milne and Ken Brown (Milne, 2012; Brown, 2007).
4.1 Affine group schemes
In the previous chapter it was shown that Affk∼= Algoppk . As k is the final object and
tensor products are the products in Algoppk , we can study group objects in the category of affinek-schemes. This leads to the following definition.
4.1.1 Definition. An affine k-group scheme is a group object in the category of affine k-schemes.
4.1.2 Remark. If C is a category and (G, m, inv, e) is a group object in SetC, it means that for every objectA in C, (G(A), mA, invA, eA) is a group object in Set.
Furthermore, the functor G corresponds to a functor G0 :C → Grp such that for the forgetful functorF : Grp→ Set, G = F ◦ G0.
Thus, an affine k-group scheme is a co-representable functor that sends k-algebras to groups.
4.1.3 Example. For ak-algebra R, the functor
Ga: Algk→ Grp, R 7→ R
is an affine k-group scheme with addition as operation.
Ga(R) = R is a group under addition and Ga is co-representable as,
Ga(R) ∼= HomAlgk(k[X], R).
4.1.4 Example. For ak-algebra R, the functor
Gm: Algk → Grp, R 7→ R× (multiplicative)
is an affine k-group scheme with multiplication as operation.
R×is a multiplicative group containing all elements inR with a multiplicative inverse,
Such that by proposition 3.2.3, Gm is co-representable and
Gm(R) ∼= HomAlgk(k[X, Y ]/(XY − 1), R).
4.1.5 Example. For an algebra R over a finite field k with characteristic p ∈ Z, the functor
αp: Algk→ Set, R 7→ {r ∈ R|rp = 0}
is an affine k-group scheme with addition as the operation.
First,αp(R) is a group closed under addition. This is the case as pi = p!/(p−i)!i! = 0
mod p for all 1≤ i < p, such that for a and b ∈ R,
(a + b)p = p X i=0 p i aibp−i=ap+ p−1 X i=1 p i aibp−i+bp =ap+bp.
Now again by proposition 3.2.3,αp is co-representable as
αp(R) ={r ∈ R|rp= 0} ∼= HomAlgk(k[X]/(X
p), R).
4.1.6 Example. For ak-algebra R, the functor
µn: Algk→ Set, R 7→ {r ∈ R|rn= 1}
is an affine k-group scheme with multiplication as the operation.
First, µn(R) is a group closed under multiplication as for a and b∈ µn(R)⊂ R×,
(a· b)n=an· bn= 1· 1 = 1 =⇒ a · b ∈ µ n(R).
Furthermore,µn(R) ={r ∈ R|rn−1 = 0} so by proposition 3.2.3, µnis co-representable,
µn(R) ∼= HomAlgk(k[X]/(X
n− 1), R).
4.1.7 Example. For n ∈ N and a k-algebra R, let GLn(R) denote the group of all
invertiblen by n matrices with coefficients in R. Then GLn: Algk→ Set, R 7→ GLn(R)
is an affine k-group scheme under matrix multiplication.
AsGLn(R) is the group of invertible n by n matrices, it is the group of matrices{rij}
such that det(rij) =p with pq = 1R for someq∈ R. So
GLn(R) ={({rij}, q) ∈ Rn
2+1
| det(rij)q− 1 = 0},
again by proposition 3.2.3,GLn is co-representable and
4.1.8 Definition. Let G, H be two affine k-group schemes. A morphism of affine group schemes is a natural transformationθ : G→ H such that the following diagram commutes. G× G θ×θ // m H× H m0 G θ //H
Here m : G× G → G is the operation on G and m0 : H× H → H is the operation on H.
Equivalently, for everyA∈ obAlgk,θA: G(A)→ H(A) is a group morphism.
4.1.9 Example. The determinant defines a morphism of affine group schemes, det :GLn→ Gm.
For R ∈ Algk and A and B ∈ GLn(R), detR(AB) = detR(A) detR(B)∈ Gm(R). This
implies that detR:GLn(R)→ Gm(R) is a group morphism for all groups Gm(R).
4.1.10 Example. If G is a commutative affine k-group scheme, then for n∈ N, [n]G: G→ G, g 7→ ng,
whereg∈ G(R) for R ∈ Algk, is a morphism of affine k-group schemes. For every R∈ Algk andf, g∈ G(R), if + the operation on G(R),
[n]GR(f + g) = n(f + g) = (f + g) +· · · + (f + g) | {z } n times =f +· · · + f | {z } n times +g +· · · + g | {z } n times =nf + ng = [n]GR(f ) + [n]GR(g).
Thus [n]GR is a group morphism for everyk-algebra R.
4.1.11 Remark. Affine k-group schemes with affine group scheme morphisms form a category denoted AffGrk.
4.2 Hopf algebras
There happens to be a strong link between affine group schemes and an algebraic struc-ture called the Hopf algebra. In this section, Hopf algebras are discussed but first, another new algebraic structure is introduced.
4.2.1 Coalgebras
4.2.2 Definition. Ak-coalgebra is a vector space A over a field k with k-linear maps ∆ :A→ A ⊗ A and : A → k,
such that the following diagrams commute. A ∆ // ∆ A⊗ A (idA,∆) (Coassociative) A⊗ A (∆,idA) //A⊗ A ⊗ A A ∆ // ∆ idA ** A⊗ A (idG,) (Counit) A⊗ A (,idA) //k⊗ A ∼=A ∼=A⊗ k
4.2.3 Definition. If (A, ∆A, A) and (B, ∆B, B) are k-coalgebras, then the k-linear map
g : A→ B is a coalgebra morphism if the following diagram commutes.
A ∆A g //B ∆B A⊗ A g⊗g //B⊗ B
4.2.4 Example. Let k be a field, then for an arbitrary set S, a vector space C = kS over
fieldk can be constructed with addition and scalar multiplication as defined in example 3.1.7. Now as this vector space has basisS, k-linear maps are completely determined by where the basis elements x∈ S are sent. Therefore, the k-linear maps
∆ :C→ C ⊗ C, x 7→ x ⊗ x and : C → k, x 7→ 1 are well defined. Now ∆ is coassociative and is a counit as for x∈ S,
(id⊗ ∆)(∆(x)) = x ⊗ (x ⊗ x) = (x ⊗ x) ⊗ x = (∆ ⊗ id)(∆(x)) and (⊗ id)(∆(x)) = 1 ⊗ x and x ⊗ 1 = (id ⊗ )(∆(x)).
From which we can conclude that (kS, ∆, ) is a k-algebra.
4.2.5 Definition. A k-coalgebra (C, ∆, ) is cocommutative if the following diagram commutes. C ∆ idC //C ∆ (Cocommutative) C⊗ C swap //C⊗ C 4.2.6 Hopf Algebras
The main reason the coalgebras over a field are introduced is to structurize some of the things discussed later. One of these things are the Hopf algebras, an algebraic structure that is defined below and is intuitively both an algebra and a coalgebra.
4.2.7 Definition. A k−Hopf algebra is ak-algebra A with k-algebra morphisms ∆ :A→ A ⊗ A, : A → k and S : A → A,
commutes. A⊗ A m //Aoo m A⊗ A A⊗ A (id⊗S) OO A ∆ oo ∆ // (e◦) OO A⊗ A (S⊗id) (Inversion) OO
4.2.8 Example. For a commutative group (G,·) and a field k, consider the the group algebra kG from example 3.1.7. It was shown that kG is a k-algebra with the corre-sponding k-linear maps mkG and ekG. As any group has an underlying set, by example
4.2.4,kG is also a k-coalgebra with the k-linear maps ∆kG andkG. Now these last two
maps arek-algebra morphisms as kG⊗kkG is a k-algebra and for f, g∈ G,
∆◦ mkG(f ⊗ g) = ∆(f · g) = (f · g) ⊗ (f · g)
= (f⊗ f) · (g ⊗ g) = mkG⊗kkG◦ (∆ ⊗ ∆)(f ⊗ g)
and
◦ mkG(f⊗ g) = (f · g) = 1 = mk(1⊗ 1) = mk◦ ( ⊗ )(f ⊗ g).
Now finally, we can construct the mapS : kG→ kG, g 7→ g−1. This map is a morphism
of k-algebras as G is commutative such that,
S◦ mkG(f⊗ g) = S(f · g) = (f · g)−1 =g−1· f−1=f−1· g−1=mkG◦ (S ⊗ S)(f ⊗ g),
and S is an inversion as
m◦ (S ⊗ id)(g ⊗ g) = m(g−1⊗ g) = g−1· g = 1 = g · g−1 =m◦ (id ⊗ S)(g ⊗ g). Thus (kG, m, e, ∆, , S) is a k-Hopf algebra.
4.2.9 Example. For a finite group (G,·) and a field k, Hom(G, k) is a k-algebra. It happens to be that this is even ak-Hopf algebra.
As Hom(G, k) = kG=Q
g∈Gk, we can define basis elements eg ∈ kGsuch thateg = 1
at thegth position and 0 everywhere else. Now as G is finite,
kG⊗ kG= Hom(G, k)⊗ Hom(G, k) ∼= Hom(G× G, k),
such that maps fromkGtokG⊗kGare isomorphic to maps from Hom(G, k) to Hom(G×
G, k). We can construct the k-algebra morphism ∆ : Hom(G, k)→ Hom(G × G, k), f 7→ f◦ m for all f ∈ Hom(G, k) from the group operation m : G × G → G. As all elements in Hom(G, k) are linear combinations of basis elements eg, and as eg = em(f,h) if and
only if m(f, h) = g∈ G, we get the k-algebra morphism ∆ :kG→ kG⊗ kG, eg 7→
X
{h,f ∈G|h·f =g}
This morphism is coassociative as (∆⊗ id) ◦ ∆(eg) = X {h,f ∈G|h·f =g} (∆⊗ id)(eh⊗ ef) = X {h,f ∈G|h·f =g} ( X {i,j∈G|i·j=h} (ei⊗ ej)⊗ ef) = X {i,j,f ∈G|(i·j)·f =g} ((ei⊗ ej)⊗ ef) = X {i,j,f ∈G|i·(j·f )=g} (ei⊗ (ej⊗ ef) = X {i,k∈G|i·k=g} (ei⊗ ( X {j,f ∈G|j·f =k} (ej⊗ ef)) = X {i,k∈G|i·k=g}
(∆⊗ id)(ei⊗ ek) = (∆⊗ id) ◦ ∆(eg)
Similarly, we get morphisms,
: kG→ k, eg 7→ ( 1 ifg = 1 0 ifg6= 1, S : kG → kG, e g 7→ eg−1,
that are a counit () and an inversion (S) making kG ak-Hopf algebra.
4.2.10 Definition. Fork-Hopf algebras (A, mA, eA, ∆A, A, SA) and (B, mB, eB, ∆B, B, SB).
A morphism of Hopf algebrasf : A→ B is a morphism of algebras with respect to thek-algebras (A, mA, eA) and (B, mB, eB), and a morphism of coalgebras with respect
to thek-coalgebras (A, ∆A, A) and (B, ∆B, B).
4.2.11 Example. For the field Fp withp∈ Z≥0, then for any Hopf algebra A over Fp,
F rA:A→ A, a 7→ ap
is a morphism of Hopf Algebras called the Frobenius morphism.
To prove that the Frobenius morphism is a morphism of Hopf Algebras, we simply check the four corresponding properties. These properties hold as for s∈ Fp, a, a0 ∈ A:
• F rA(a + a0) = (a + a0)p =ap+a0p=F rA(a) + F rA(a0) by the binomial theorem,
shown in example 4.1.5
• F rA(sa) = (sa)p =spap=sap =sF rA(a)
• F rA(aa0) = (aa0)p=apa0p=F rA(a)F rA(a0)
• F rA⊗A(a⊗ a0) = (a⊗ a0)p =ap⊗ a0p=F rA(a)⊗ F rA(a0)
4.2.12 Remark. The Hopf algebras over a fieldk form a category Hopfkwith as objects the Hopf algebras over fieldk and as morphisms the Hopf algebra morphisms.
4.3 Correspondence of Hopf algebras and affine group
schemes
Due to the equivalence between the category of affine k-schemes and the opposite cat-egory of k-algebras, an affine k-group scheme is simply a group object in Algoppk . The following lemma builds on this alternative definition of affine k-group schemes.
4.3.1 Lemma. The opposite category ofk-Hopf algebras is equivalent to the category of affinek-group schemes.
Proof. In the previous chapter, it was shown that tensor products and initial objects in Algk are products and final objects in Algoppk . Combining this to what is discussed about affinek-group schemes and opposite categories lead to the following equivalences: • ∆ : A → A ⊗ A is a coassociative morphism in Algkif and only if ∆ :A⊗ A → A
is an associative morphism in Algoppk
• : A → k is a counit in Algk if and only if : k→ A is a unit in Alg opp k
• S : A → A is an inversion in Algk if and only ifS : A→ A is an inverse in Algoppk
Hereby, (A, ∆, , S) is a k-Hopf algebra if and only if (A, ∆, , S) a group object in Algoppk .
4.3.2 Remark. For an affine k-group scheme G and a k-algebra A, there is a k-algebra R such that G(A) = Hom(R, A) and
G(A)× G(A) = Hom(R, A) × Hom(R, A) = Hom(R ⊗ R, A).
So by Yoneda, for a fixed objectA, there is a morphism mA: G(A)×G(A) → G(A) if and
only if there is ak-algebra morphism ∆ : R→ R ⊗ R such that mA=h∆(A). Similarly,
ask ∼= Hom(k, A), eA:k→ G(A) corresponds to a k-algebra morphism : R → k with
eA=h(A) and invA: G(A) → G(A) corresponds to a k-algebra morphism S : R → R
such thatinvA=hS(A).
As affine group schemes and Hopf algebras are now clearly linked, it is possible to construct the Hopf algebras corresponding to some of the affine group schemes mentioned in the previous section.
4.3.3 Example. To find the k-Hopf algebra corresponding to the affine k-group scheme Ga, we first need to obtain morphisms ∆, and S corresponding to the natural
trans-formationsm, e and inv of Ga.
To construct these morphisms, consider a k-algebra A and f, g∈ HomAlgk(k[X], A).
This gives an induced map (f⊗g) ∈ HomAlgk(k[X]⊗k[X], A), such that (f ⊗g)(X ⊗1) =
f (X) and (f ⊗ g)(1 ⊗ X) = g(X). Now Ga is an affinek-group scheme with regards to
the natural transformationm defined such that for all A∈ Algk,
In the remark above, it was shown that there is a morphism ∆ : k[X] → k[X] ⊗ k[X] such thatmA=h∆(A). Thus for f, g∈ HomAlgk(k[X], A) and their induced map f⊗ g,
mA(f⊗ g)(X) = f(X) + g(X) = (f ⊗ g)(X ⊗ 1 + 1 ⊗ X) = (f ⊗ g) ◦ ∆(X).
This means that the morphism,
∆ :k[X]→ k[X] ⊗ k[X], X 7→ X ⊗ 1 + 1 ⊗ X,
corresponds to the natural transformation m of Ga. Similarly, the morphisms and S
given by,
: k[X]→ k, X 7→ 0 and S : k[X] → k[X], X 7→ −X, correspond toe and inv of Ga.
Now lets check that these newly constructed k-algebra morphisms indeed make the k-algebra k[X] into a k-Hopf algebra.
Coassociativity of ∆ holds as (∆⊗ id) ◦ ∆(X) = (∆ ⊗ id)(X ⊗ 1 + 1 ⊗ X) = (X⊗ 1 + 1 ⊗ X) ⊗ 1 + 1 ⊗ 1 ⊗ X =X⊗ 1 ⊗ 1 + 1 ⊗ X ⊗ 1 + 1 ⊗ 1 ⊗ X =X⊗ 1 ⊗ 1 + 1 ⊗ (X ⊗ 1 + 1 ⊗ X) = (id⊗ ∆)(X ⊗ 1 + 1 ⊗ X) = (id⊗ ∆) ◦ ∆(X), and the morphism : k[X]→ k is a counit as,
(⊗ id) ◦ ∆(X) = ( ⊗ id)(X ⊗ 1 + 1 ⊗ X) = 1 ⊗ X ∼=X, (id⊗ ) ◦ ∆(X) = (id ⊗ )(X ⊗ 1 + 1 ⊗ X) = X ⊗ 1 ∼=X. All that is left to show now is that S : k[X]→ k[X] is an inversion,
n◦ (id ⊗ S) ◦ ∆(X) = n ◦ (id ⊗ S)(X ⊗ 1 + 1 ⊗ X) = n(X ⊗ 1 + 1 ⊗ (−X)) =X + (−X) = (X) = (−X) + X = n ◦ (S ⊗ id) ◦ ∆(X). So we can conclude that (k[X], ∆, , S) is the k-Hopf Algebra corresponding to the affine k-group scheme Ga.
4.3.4 Example. Gm corresponds to a k-Hopf Algebra k[X, X−1] with k-algebra
mor-phisms:
∆ :k[X, X−1]→ k[X, X−1]⊗ k[X, X−1], X 7→ (X ⊗ 1)(1 ⊗ X) : k[X, X−1]→ k, X 7→ 1
obtained in a similar fashion as in example 4.3.3. Coassociativity of ∆ : k[X, X−1] →
k[X, X−1]⊗ k[X, X−1] holds as:
(∆⊗ id) ◦ ∆(X) = (∆ ⊗ id)(X ⊗ X) = (X ⊗ X) ⊗ X)
=X⊗ (X ⊗ X) = (id ⊗ ∆)(X ⊗ X) = (id ⊗ ∆) ◦ ∆(X). The morphism : k[X, X−1]→ k is a counit as:
(⊗ id) ◦ ∆(X) = ( ⊗ id)(X ⊗ X) = (1 ⊗ X) ∼=X ∼= (X⊗ 1) = (id ⊗ ) ◦ ∆(X). Furthermore,S : k[X, X−1]→ k[X, X−1] satisfies the inversion property as,
m◦ (id ⊗ S) ◦ ∆(X) = m ◦ (id ⊗ S)(X ⊗ X) = m((X ⊗ X−1))
=XX−1 =(X) = X−1X = m◦ (S ⊗ id) ◦ ∆(X). Therefore, (k[X, X−1], ∆, , S) is the k-Hopf algebra corresponding to Gm.
4.3.5 Example. As for all k-algebras A, µn(A) ⊂ Gm(A) and both form a group under
multiplication,µn corresponds to thek-Hopf algebra k[X]/(Xn− 1) with morphisms:
∆ :k[X]/(Xn− 1) → k[X]/(Xn− 1) ⊗ k[X]/(Xn− 1), X 7→ (X ⊗ 1)(1 ⊗ X)
: k[X]/(Xn− 1) → k, X 7→ 1
S : k[X]/(Xn− 1) → k[X]/(Xn− 1), X 7→ X−1
∆ is coassociative, is counit and S is an inversion as shown in example 4.3.4.
4.3.6 Example. For algebras R over a finite field k with characteristic p, as αp(R) ⊂
Ga(R) and both form a group under addition, αp corresponds to the k-Hopf algebra
k[X]/(Xp) with morphisms:
∆ :k[X]/(Xp)→ k[X]/(Xp)⊗ k[X]/(Xp), X7→ X ⊗ 1 + 1 ⊗ X
: k[X]/(Xp)→ k, X 7→ 0
S : k[X]/(Xp)→ k[X]/(Xp), X 7→ −X
∆ is coassociative, is counit and S is an inversion as shown in example 4.3.3.
4.3.7 Remark. As Hopf algebras correspond to affine group schemes, we know that for a commutative group G, there is an affine k-group scheme denoted µG, that can be
obtained from the group algebrakG. Similarly, if the group G is finite, then there is an affine k-group scheme corresponding to the k-Hopf algebra kG, usually denoted G.
5 Cartier Duality
In the final chapter of this thesis, Cartier duality is discussed. First, duality of vector spaces and some of their properties are mentioned. This is used to define a dual functor which happens to be an equivalence of categories for both finite vector spaces as finitely generated commutative Hopf algebras. Through this functor, the Cartier dual of a finitely generated commutative Hopf algebra is obtained. To conclude the chapter, two well known examples of Cartier duality are given. In this final chapter, the notes from James Milne and Richard Pink, and William C. Waterhouse’s book about affine group schemes were used (Milne, 2012; Pink, 2004; Waterhouse, 1979).
5.1 Dual vector spaces
5.1.1 Definition. For a vector space V over a field k, the dual vector space is the naturally induced vector spaceV∨ := Hom(V, k).
5.1.2 Lemma. IfV is a finitely generated vector space over a field k with basis v1, . . . , vn
for some n∈ N, then the dual vector space V∨ has a basis v∨
1, . . . , v∨n, such that
v∨i (vj) =
(
1 if i = j 0 if i6= j. Proof. For f ∈ Hom(V, k) and v =Pn
i=1aivi ∈ V with ai ∈ k for 1 ≤ i ≤ n, v ∨ i (v) = ai such that f (v) = f n X i=1 aivi = n X i=1 f (vi)ai = Xn i=1 f (vi)v∨i (v). So f =Pn i f (vi)v ∨ i = Pn i biv ∨
i , where bi =f (vi)∈ k for 1 ≤ i ≤ n. Thus every k-linear
map from V to k can be constructed through linear combinations of v∨
1, . . . v∨n with
coefficients in k. To show linearly independence, assume that f =Pn
i=1bivi∨= 0. Then
asbi =f (vi) for 1≤ i ≤ n, bi = 0 for 1≤ i ≤ n. Thus v1∨, . . . , vn∨ is a basis of V∨.
5.1.3 Remark. For finitely generated vector spaces V and W over a field k, a k-linear mapf : V → W induces a new k-linear map f∨ :W∨ → V∨ such thatf∨(g) = g◦ f for all g∈ W∨. This morphism can be seen as the transpose of f .
For basis v1, . . . , vn of V and w1, . . . , wm of W with n, m∈ N, V ∼=kn and W ∼=km
such that a k-linear map f : V → W is equivalent to a transformation from kn tokm.
So ak-linear map f can be expressed by a matrix (aij)i,j with 1≤ i ≤ n, 1 ≤ j ≤ m and
and for v =Pn
i=1bivi ∈ V with bi coefficients in k, for a basis element w∨r,
f∨(w∨r)(v) = wr∨◦ f(v) = wr∨( m X j=1 ( n X i=1 (aijbi)wj) = n X i=1 airbi= n X i=1 airv∨i (v).
Hereby,f (vr) =Pmj=1arjwj if and only if f∨(w∨r) =
Pn
i=1airv∨i.
5.1.4 Lemma. Let FVeck denote the category of finitely generated vector spaces over
a field k. For V ∈ FVeck, the dual functor
(−)∨ : FVecoppk → FVeck, V 7→ V∨ and f 7→ f∨
is an equivalence of categories and (V∨)∨ ∼=V for all V ∈ FVec k.
Proof. ForV ∈ FVeckandv∈ V , consider the natural transformation ϕ : id 7→ ((−)∨)∨
defined by:
ϕV :V → (V∨)∨, v7→ fv,
wherefv is the k-linear map fv :V∨ → k such that for every g ∈ V∨, fv(g) = g(v).
Then ϕV is injective as for v∈ V , if v ∈ ker(ϕV), then
0 =ϕ(v) = fv(g) = g(v) for all g∈ V∨,
which is only the case if v = 0. Surjectivity of ϕ holds as dim(V ) = dim(V∨) =
dim((V∨)∨) (this is a result of lemma 5.1.2). Thusϕ is a natural isomorphism such that V ∼= (V∨)∨ for all V ∈ FVeck.
5.1.5 Lemma. For finitely generated vector spaces V and W over a field k, V∨⊗ W∨ ∼= (V ⊗ W )∨.
Proof. For f ∈ Hom(V, k) and g ∈ Hom(W, k), consider the k-linear map ψ : V∨⊗ W∨ → (V ⊗ W )∨, f⊗ g 7→ fg where fg(v ⊗ w) = f(v)g(w).
We will now show thatψ is an isomorphism. If{vi}1≤i≤n is a basis ofV and{wj}1≤j≤m
is a basis of W for n, m∈ Z, then {v∨
i ⊗ w∨j} is a basis for V∨⊗ W∨ and {(vi⊗ wj)∨}
is a basis for (V ⊗ W )∨. Now for 1≤ i, p ≤ n and 1 ≤ j, q ≤ m,
ψ(v∨i ⊗ wj∨)(vp⊗ wq) =vi∨(vp)w∨j(wq) =
(
1 if (i, j) = (p, q) 0 else.
Thusψ(vi∨⊗ w∨
j) = (vi⊗ wj)∨ such that all basis elements ofV∨⊗ W∨ get sent to basis
elements of (V⊗W )∨. Now as dim(V∨⊗W∨) = dim((V⊗W )∨),ψ is an isomorphism.
5.1.6 Remark. For a field k and f ∈ k∨, as f is completely determined by where 1 is
sent,
5.2 Cartier Duality
5.2.1 Duality of Algebras
In the third and fourth chapter, when algebra’s and coalgebra’s were introduced, it was mentioned that if V is a vector space, then for a morphism m : V ⊗ V → V to be associative and a morphism ∆ :V → V ⊗ V to be coassociative, the following diagrams should commute. V ⊗ V ⊗ V (id⊗m) // (m⊗id) V ⊗ V m (Associative) V ⊗ V m //V V ⊗ V ⊗ V oo (id⊗∆) V ⊗ V V ⊗ V (∆⊗id) OO V ∆ oo ∆ (Coassociative) OO
In the previous section, it was shown that (V ⊗ W )∨ ∼=V∨⊗ W∨ andk∨ ∼=k. As every morphism f ∈ Hom(V, W ) has an induced morphism f∨ ∈ Hom(W∨, V∨), taking the
dual of the above diagrams give the following two commutative diagrams for thek-linear maps m∨ :V∨ → V∨⊗ V∨ and ∆∨:V∨⊗ V∨ → V∨. V∨⊗ V∨⊗ V∨ oo (id⊗m∨) V∨⊗ V∨ V∨⊗ V∨ (m∨⊗id) OO V∨ m∨ oo m∨ (Coassociative) OO V ∨⊗ V∨⊗ V∨ (id⊗∆∨) // (∆∨⊗id) V∨⊗ V∨ ∆∨ (Associative) V∨⊗ V∨ ∆∨ // V∨
Comparing these two diagrams with the ones above, it becomes clear that the mapm∨: V∨ → V∨⊗ V∨ is coassociative is and only ifm is associative and ∆∨ :V∨⊗ V∨ → V∨ is associative if and only if ∆ is coassociative. Listed below are some structures that interchange in a similar way by the dual functorV 7→ V∨.
m : V ⊗ V → V (associative) ⇐⇒ m∨ :V∨ → V∨⊗ V∨ (coassociative) e : k→ V (unit) ⇐⇒ e∨ :V∨ → k (counit)
S : V → V (inversion) ⇐⇒ S∨:V∨ → V∨ (inversion)
∆ :V → V ⊗ V (coassociative) ⇐⇒ ∆∨ :V∨⊗ V∨ → V∨ (associative) : V → k (counit) ⇐⇒ ∨ :k→ V∨ (unit)
Remark From the previous list, it can be concluded that if (A, m, e) is a finitely generatedk-algebra, then (A∨, m∨, e∨) is a finitely generatedk-coalgebra and if (A, m, e) is commutative, then (A∨, m∨, e∨) cocommutative. Similarly, if (C, ∆, ) is a finitely
generatedk-coalgebra, then (C∨, ∆∨, ∨) is a finitely generatedk-algebra and if (C, ∆, ) is cocommutative, then (C∨, ∆∨, ∨) is commutative.
5.2.2 Cartier Dual
5.2.3 Theorem. Let CFHopfk denote the category of finitely generated commutative cocommutative Hopf algebras over a fieldk. Then for A∈ CFHopfoppk ,
(−)∨ : CFHopfoppk → CFHopfk, A7→ A∨
is an equivalence of categories and the k-Hopf algebra (A∨, ∆∨, ∨, m∨, e∨, S∨) is called
the Cartier dual of (A, m, e, ∆, , S).
Proof. As (−)∨ : FVecoppk → FVeck is an equivalence of categories, it is sufficient to
show that (−)∨ maps finitely generated commutative cocommutative k-Hopf algebras
to finitely generated commutative cocommutative k-Hopf algebras and Hopf algebra morphisms to Hopf algebra morphisms.
First, it will be shown that (A∨, ∆∨, ∨, m∨, e∨, S∨) is indeed a commutative cocom-mutative k-Hopf algebra. It was already shown that (A∨, m∨, e∨) is a cocommutative k-coalgebra and that (A∨, ∆∨, ∨) is a commutative k-algebra in the previous section.
On the list on the previous page, it is also shown that the morphism S∨ satisfies the inversion property such that the only thing left to show is that m∨, e∨ and S∨ are k-algebra morphisms. This will be done in the case ofm∨ andS∨, the proof ofe∨ is done
similarly. Recall that ∆ and S are k-algebra morphisms and that A is cocommutative such that the following diagrams commute.
A⊗ A ∆⊗∆ // m (A⊗ A) ⊗ (A ⊗ A) m⊗m A ∆ //A⊗ A A ∆ // S A⊗ A S⊗S A ∆ //A⊗ A
Applying the dual functor to these diagrams give the commutative diagrams below,
A∨⊗ A∨ oo (∆⊗∆)∨ (A∨⊗ A∨)⊗ (A∨⊗ A∨) A∨ m∨ OO A∨⊗ A∨ ∆∨ oo (m⊗m)∨ OO A ∨ oo ∆∨ A∨⊗ A∨ A∨ S∨ OO A∨⊗ A∨ ∆∨ oo S∨⊗S∨ OO
As (∆⊗∆)∨= ∆∨⊗∆∨and (m⊗m)∨ =m∨⊗m∨, we see thatm∨ andS∨arek-algebra
morphisms such that (A∨, ∆∨, ∨, m∨, e∨, S∨) is a commutative cocommutative k-Hopf algebra.
Now if A, B ∈ CFHopfk and f ∈ HomCFHopfk(A, B), then the following diagrams
commute. A⊗ A f ⊗f // mA B⊗ B mB A f //B A f // ∆A B ∆B A⊗ A f ⊗f //B⊗ B
Applying the dual functor to these diagrams gives the following commutative diagrams. A∨⊗ A∨ oo (f ⊗f )∨ B∨⊗ B∨ A∨ m∨ A OO B∨ m∨ B OO f∨ oo A∨ f B∨ ∨ oo A∨⊗ A∨ ∆∨ A OO B∨⊗ B∨ (f ⊗f )∨ oo ∆∨ B OO
The commutivity of these diagrams precicely define f∨ to be a morphism of k-Hopf
algebras.
5.2.4 Remark. Just like for finite vector spaces, a result of the above theorem is that for A∈ CFHopfk,
(A∨)∨∼=A.
5.2.5 Remark. As every finitely generated commutative cocommutativek-Hopf algebra A corresponds to a finitely generated commutative affine k-group scheme G, there is an associated dual affinek-group scheme corresponding to A∨, called the Cartier dual of G and denoted D(G).
We will now compute the Cartier dual of the affinek-group schemes αp and Z/nZ.
5.2.6 Example. Ifk is a finite field with characteristic p∈ N, then from example 4.3.6, we know thatαpis an affinek-group scheme corresponding to the k-Hopf algebra k[X]/(Xp)
with basis elements{Xi}
0≤i<p and morphisms:
m : k[X]/(Xp)⊗ k[X]/(Xp)→ k[X]/(Xp), Xi⊗ Xj 7→ Xi+j e : k→ k[X]/(Xp), 17→ X0 ∆ :k[X]/(Xp)→ k[X]/(Xp)⊗ k[X]/(Xp), Xi 7→ i X j=0 i j (Xj ⊗ Xi−j) : k[X]/(Xp)→ k, X i 7→ ( 1 ifi = 1 0 else S : k[X]/(Xp)→ k[X]/(Xp), Xi 7→ (−X)i
As this is a finitely generated commutative cocommutativek-Hopf algebra, we can con-struct its Cartier dual. Let{ei}0≤i<p be the basis of the dual vector space (k[X]/(Xp))∨
corresponding to the basis {Xi}
0≤i<p. To find the Cartier dual of this k-Hopf algebra,
we will construct the dual of each morphism.
For the morphism m∨ : (k[X]/(Xp))∨ → (k[X]/(Xp))∨ ⊗ (k[X]/(Xp))∨ and basis
elementei,m∨(ei) =ei◦ m such that for basis elements Xm, Xn∈ k[X]/Xp,
(ei◦ m)(Xm⊗ Xn) =ei(Xm+n) =
(
1 ifm + n = i 0 else.
AsXm and Xn were arbitrary basis elements ofk[X]/(Xp), it follows that m∨(ei) = i X j=0 ej⊗ ei−j.
Now for ∆∨ : (k[X]/(Xp))∨⊗ (k[X]/(Xp))∨ → (k[X]/(Xp))∨, we know ∆∨(e
i⊗ ej) = (ei⊗ ej)◦ ∆ such that ∆∨(ei⊗ ej) = (ei⊗ ej)◦ ∆(Xm) = (ei⊗ ej)(X⊗ 1 + 1 ⊗ X)m = (ei⊗ ej)( m X n=0 m n Xn⊗ Xm−n) = m X n=0 m n ei(Xn)⊗ ej(Xm−n) = ( i+j i ifi + j = m 0 else.
So as Xm was an arbitrary basis element ofk[X]/(Xp),
∆∨(ei⊗ ej) =
i + j i
ei+j.
In a similar fashion, we obtaine∨ and∨ such that,
e∨ : (k[X]/(Xp))∨→ k, ei 7→ ( 1 ifi = 1 0 else and ∨ :k → (k[X]/(Xp))∨, 17→ e0.
Now from the map ∆∨, it follows thatei = (e1)i/i! for all 1≤ i ≤ p. So we can express
every basis element in terms of e1 and get (k[X]/(Xp))∨ ∼= k[e1]/(ep1). This also gives
us an easy definition of the map S∨ as,
S∨ : (k[X]/(Xp))∨→ (k[X]/(Xp))∨
, e1 7→ −e1.
If we do a change of variablese1 :=X, it follows that
(k[X]/(Xp))∨, ∆∨, ∨, m∨, e∨, S∨) = (k[X]/(Xp), m, e, ∆, , S).
Which implies that thek-Hopf algebra corresponding to αp is its own Cartier dual and
thus thatD(αp) =αp.
thek-Hopf algebra corresponding to the finite affine k-group scheme Z/nZ is kZ/nZ with: m : kZ/nZ⊗ kZ/nZ → kZ/nZ, e a⊗ eb 7→ ( ea+b ifea=eb 0 ifea6= eb e : k→ kZ/nZ, 17→ X a∈Z/nZ ea ∆ :kZ/nZ→ kZ/nZ⊗ kZ/nZ, e a7→ X {b,c∈Z/nZ|b·c=a} eb⊗ ec : kZ/nZ→ k, e a7→ ( 1 ifa = 1 0 else S : kZ/nZ→ kZ/nZ, e a7→ e−a
Now if the Cartier dual ofkZ/nZ is taken with corresponding dual basis{e∨
a}a∈Z/nZ, the
following morphisms are optained:
m∨ : (kZ/nZ)∨ → (kZ/nZ)∨⊗ (kZ/nZ)∨, e∨ a 7→ e ∨ a⊗ e ∨ a e∨ : (kZ/nZ)∨ → k, e∨ a 7→ 1 ∆∨ : (kZ/nZ)∨⊗ (kZ/nZ)∨ → (kZ/nZ)∨, e∨ a ⊗ e∨b 7→ e∨a+b ∨ :k→ (kZ/nZ)∨, 17→ e∨ 0 S∨ : (kZ/nZ)∨ → (kZ/nZ)∨, e∨ a 7→ e ∨ −a Now ase∨
a = (e∨1)a,e1∨ generates all basis elementse∨a such that fromX := e∨1, it follows
that (kZ/nZ)∨ ∼=k[X]/(Xn− 1). As the morphisms above correspond to those making
k[X]/(Xn− 1) into the k-Hopf algebra in example 4.3.5, (kZ/nZ)∨ ∼=k[X]/(Xn− 1) as
k-Hopf algebras. Now as the k-Hopf algebra kZ/nZ corresponds to the affine k-group
scheme Z/nZ and k[X]/(Xn− 1) corresponds to the affine k-group scheme µn, it follows
thatD(Z/nZ) ∼=µn as affinek-group schemes.
The two examples given above are the main examples of Cartier duality and play an important role in the classification of finite affine group schemes over a field. Cartier duality in general and these examples specifically also increase intuition of what affine group schemes are as they require the varification and construction of the morphisms that define them.
6 Conclusion
To get an understanding of affine group schemes, it is necessary to first lay a categorical basis. This was done in the first chapter where categories and functors were discussed. Besides this categorical basis, some new algebraic structures needed to be defined and a different view towards groups needed to be obtained. These preparations give rise to a new mathematical structure that is the affine schemes. From this point on, a start can be made to understanding affine group schemes. Though there are a lot of examples of affine group schemes, they are not always easy to obtain. A major tool to find and classify affine group schemes is through an algebraic structure in close correspondence to them called Hopf algebras. This corresondence gives a different insight into affine group schemes and understanding this correspondence is the main focus of this project.
In the last chapter, Cartier duality of finitely generated commutative cocommutative Hopf algebras was covered. Cartier duality is the analogue to Pontryagin duality of groups and plays an important role in the classification of finite affine group schemes. This classification could be an interesting follow up project.
Through basic category theory, an entry level notion of what affine group schemes are has been obtained. Though abstract, it does not involve many difficult proofs making it readable for a bachelor level mathematician. As affine group schemes are a quite advanced subject within algebra, it is interesting how a basic knowledge about them can be obtained in such a short time and with a relatively small mathematical basis. The goal of this bachelorproject was to gain a basic understanding of what affine group schemes are and this is assumed to be achieved.
7 Popular summary
An algebraic structure is a set of elements with one or more operations on it that satisfy some properties. To give an example of such an algebraic structure, consider the set of all rational numbers. If we add the operation addition to this set, it becomes an algebraic structure called a group. Adding a second scalar to our group in the form of multiplication gives us a new algebraic structure called a ring. In the case of the set of all rational numbers, it happens that we can continue this process such that it can also be viewed as a field, a vector space or an algebra over a field, three other well-known algebraic structures.
It is sometimes the case that we want to give a set, that already embodies a certain mathematical structure, a group structure. In this case, these two structures need to coincide in some way. When this is the case for some group operation, we call this a group object with respect to the corresponding group operation. These group objects can be constructed on a wide range of mathematical structures and in this thesis, the main focus is on the group objects on one of these mathematical structures called the affine schemes.
Before we talk about affine schemes, it is important to obtain a different perspective on what maps are. A map from a set of elements (A) to another set of elements (B) sends every element in the setA to an element in the set B. In general, this can be done in many different ways and all these different mappings also form a set. To summarize, we can construct a set containing all maps from setA to set B.
If we now consider A and B to be algebras over a field, then like before, there is a set of all maps from A to B that preserve the algebra structure. An affine scheme is a kind of mapping, that sends an algebra over a field B, to the set of all maps from a fixed algebra over a field A to B. This affine scheme is in some way “represented” by the fixed algebraA.
The affine schemes are mathematical structures and for some of them, they are group objects. Thus, in some cases, a group structure can be constructed on an affine scheme. These group objects are called affine group schemes and intuitively, they are mappings that send an algebra over a field B, to the set of all maps from a fixed algebra over a field A to B such that this set of maps has a group structure. These affine group schemes are the main focus of this thesis. There correspondence to another algebraic structure called the Hopf algebra is discussed thoroughly and a way of classifying affine group schemes is given.
References
Awodey, S. (2010). Category theory (Second ed.). Oxford university press.
Brown, K. (2007). Hopf algebras.
(http://www.maths.gla.ac.uk/ kab/Hopf%20lects%201-8.pdf) Milne, J. S. (2012). Basic theory of affine group schemes.
(www.jmilne.org/math/)
Pink, R. (2004). Finite group schemes.
(https://people.math.ethz.ch/ pink/ftp/FGS/CompleteNotes.pdf) Taelman, L. (2017). Modules and categories.
(https://staff.fnwi.uva.nl/l.d.j.taelman/ca.pdf) van Oosten, J. (2002). Basic category theory.
(https://www.staff.science.uu.nl/ ooste110/syllabi/catsmoeder.pdf)
