1 Recall. Algebraic Groups Seminar Andrei Frimu Talk 4: Cartier Duality

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1 1 ecall Assume we have a locally small category C which admits finite products and has a final object, i.e. an object so that for every Z Ob(C), there exists a unique morphism Z. Note that two morphisms f, g Z G can then be identified uniquely with a morphism (f, g) : Z. Definition. A commutative group object in C is a pair consisting of an object G Ob(C) and a morphism : G such that for every object Z Ob(C), the map Mor(Z, G) Mor(Z, G) Mor(Z, G) sending (g, g ) (g, g ) defines a commutative group. There is an alternate description of commutative group objects which sheds more light on what associativity, commutativity, existence of identity and inverses means. Before we state the proposition, note that the existence of products implies the existence of a morphism σ : interchanging the two factors. Proposition. An object G Ob(C) and a morphism G constitute a commutative group object in C if and only if the following properties hold: Associativity: The following diagram is commutative: G id id G. Commutativity: The following diagram is commutative: G σ Identity element: There exists a morphism e : G such that the following diagram commutes G e id pr 2 G Inverses: There exists a morphism i : G G such that the following diagram commutes id i G diag e G where e is the morphism from the previous bullet point. 1

2 2 Affine Group Schemes Let ings be the category of commutative noetherian rings with 1. The morphisms of this category are ring homomorphisms preserving the identity. ecall that there is an (anti-)equivalence of categories between ings and the category Aff.Sch of affine schemes. ecall that an object A of ings together with a morphism A constitutes a unitary -algebra. Note that since A is a ring and the multiplication A A A is -bilinear, it induces an -module homomorphism : A A A as follows A A A A A This allows us to (equivalently) view a unitary -algebra A as an -module with two homomorphisms of -modules e A A A, satisfying (a a ) = (a a) (corresponding to multiplication in A being commutative) and (a (a a )) = ((a a ) a ) (corresponding to multiplication in A being associative) and (e(1) a) = a (corresponding to a e(1) = a). Denote the category of -algebras by -Alg. Then the (anti-)equivalence of categories ings and Aff.Sch restricts to an (anti-)equivalence of categories -Alg and Aff. -Sch. Definition. An affine commutative group scheme over Spec is a commutative group object in the category of affine schemes over Spec. Let G = Spec A be an affine commutative scheme over Spec. Then in particular A is a unitary -algebra. Combining the categorical definitions for unitary -algebras and commutative group object, we obtain the following diagram of -module homomorphisms: ɛ e A A A, m i The axioms satisfied by all the morphisms above turn A into a Hopf -algebra. Definition. A homomorphism of group schemes Φ : G H over Spec is a morphism in Aff. -Sch, such that the induced morphism Mor(Z, G) Mor(Z, H) is a homomorphism of groups for all Z Ob(Aff. -Sch). If G = Spec A and H = Spec B, this morphism corresponds to a homomorphism of -modules φ : B A such 2

3 that the following diagram commutes: ɛ A A A A A e A m A id i A φ φ φ ɛ B B B B B. e B m B i B Definition. The sum of two homomorphisms Φ, Ψ : G H is defined by the commutative diagram G diag Φ+Ψ H H Φ Φ H H 3 n,k and α p, Let s step aside from the course of the notes to define two objects of interest later. ecall that the multiplicative group over k is G m,k def = Spec k[t, T 1 ]. This is a group scheme, with co-multiplication m : k[t, T 1 ] k[t 1, T 1 1 ] k k[t 2, T 1 2 ] given by T T 1 T 2. Consider the homomorphism n id : G m,k G m,k induced by the map (on rings) k[t, T 1 ] k[t, T 1 ] sending t t n. We are interested in the fiber produce n,k = G m,k Gm,k Spec k in the diagram G m,k n id G m,k n,k Spec k where the map Spec k G m,k is induced from k[s, S 1 ] k sending S 1. For affine schemes, fiber products correspond to the tensor product of the corresponding rings, implying that n,k = Spec A, where A = k[t, T 1 ] k[s,s 1 ] k. Using T n 1 = S(1 1) = 1 S = 1 1, one concludes that A = k[t ]/(T n 1), hence n,k = Spec k[t ]/(T n 1). This is a group scheme with comultiplication m sending, just like for G m,k (but in different rings) T T 1 T 2. The other object we introduce is α n, = Spec [T ]/(T n ). This is a group scheme with comultiplication m : [T ]/(T n ) [T 1 ]/(T1 n ) [T 2 ]/(T2 n ) given by m(t ) = T T 2. Note that over a field of positive characteristic p, with n = p s, we have T ps 1 = (T 1) ps, and hence p s,k Spec k[u]/(u ps ) α p s,k, where we identified U = T 1. As a particular case, notice that we have established p,k α p,k as schemes over k. By the end of the lecture, we shall see that p,k and α p,k are not isomorphic in the category of affine commutative finite flat group schemes over Spec k. 3

4 4 Cartier Duality Assume now that G = Spec A is a group scheme, finite and flat over (i.e. that A is a locally free, finitelygenerated -module). Let A def = Hom (A, ) be the -dual of A. One can check that if we dualize the diagram following the definition of an affine commutative group scheme, one obtains the following diagram, under the identifications = and (A A) = A A : e ɛ A A A. i m The morphisms e, m,, ɛ and i satisfy the axioms of a cocommutative Hopf algebra with antipodism, and therefore G def = Spec A is an affine commutative finite flat group scheme over Spec. Definition. G is called the Cartier dual of G. If we have homomorphism Φ : G H of affine commutative finite flat group schemes over Spec corresponding to a homomorphism φ : B A, then φ induces φ : A B and thus a homomorphism of group schemes H G. Hence Cartier duality gives a contravariant functor from the category of affine group schemes to itself. This functor is additive, i.e. given Ψ, Φ : G H, then (Ψ + Φ) = Ψ + Φ. 5 Constant Group Schemes Let G be an additive finite abelian group. We want to exhibit a finite commutative group scheme associated to G. For that we take the disjoint union of G copies of the final object Spec = in the category of affine schemes over Spec. Let G (standard notation for group schemes) be G = Spec. g G Notation: Write (Spec ) g = Spec for the g-component of G. Exercise: Why is this a scheme over Spec? What is the unique morphism to the terminal object Spec? Exercise: G is finite and flat over Spec. As written, it is not obvious that this is a commutative group scheme over Spec. For that, note first that G G Spec. g,g G This follows from the fact that the product of two disjoint unions X = X α and Y = Y β of -schemes is the disjoint union of the products X α Y β (EGA, I.3.2.8) and from (Spec Spec ) Spec (EGA Chapter I.3.2.2). Define then the morphism : G by sending the component (Spec ) g,g of G G to the (Spec ) g+g component of G. 4

5 Exercise: The scheme G and the morphism define a commutative group scheme over Spec. The scheme G is also called the constant group scheme over with fiber G. Lemma. Let G def = {f : G f is a map of sets}. This is a ring, with addition and multiplication defined componentwise. The zero and the identity are the constant maps with value 0, respectively 1. Then G Spec( G ). Proof. This follows by induction from EGA I However, as it is necessary in the subsequent discussion, we can describe the isomorphisms between the two sides in the following way: As a convenience of notation, let g be a copy of the ring, and assume the g-component of G is (Spec ) g = Spec g. For every such component we have a morphism Spec g Spec( G ) induced by the ring homomorphism G g sending f to f(g) = g. This then induces a morphism φ : G Spec( G ). Conversely, we have a morphism ψ : Spec( G ) G, induced by the ring homomorphisms g G sending an element r = g to the map f : G satisfying f(g) = r and f(g ) = 0 for g g. Exercise: φ and ψ are isomorphisms, inverse to each other. In particular G r G r is Spec( G G ) and thus the map G G G is induced by the the comultiplication map m : G G G G G. Let s explicitly define this map. Let f G. From the map on components G G G defined above, one can deduce that m(f) is the map of G G sending (g, g ) to f(g + g ). The counit G is defined by ɛ(f) = f(0). The coinverse i : G G is, as expected, the map sending g f( g). Consider now the canonical basis {e g } of G given by e g : G such that e g (g) = 1 and e g (g ) = 0 for g g. Using the above, one can check that we have (e g e g ) = (e g if g = g ) and (0 otherwise) ɛ(e g ) = (1 if g = 0) and (0 otherwise) e(1) = e g g G m(e g ) = e g e g g g G i(e g ) = e g. To see what the Cartier dual of G is, let (ê g ) g G be the basis of ( G ) dual to the above basis, i.e. ê g (g ) = 1 5

6 if g = g and 0 if g g. Then the dual maps are given by (ê g ) = ê g ê g ɛ (1) = ê 0 e (ê g ) = 1 m (ê g ê g ) = ê g+g i (ê g ) = ê g. Proposition. The formulas for m and ɛ show that ( G ) is isomorphic to the group [G] as an -algebra, such that e corresponds to the usual augmentation map [G]. Example 1: G = Z/nZ Denote X def = ê 1. Then ( G ) [G] = [Z/nZ] = [X]/(X n 1). The comultiplication : ( G ) ( G ) ( G ) is (X) = X X, which we note is the same as what holds for n,. Thus (Z/nZ) n,. Dual of α p, : Assume has characteristic p and let A = [T ]/(T p ). ecall α p, = Spec A is a group scheme with comultiplication m(t ) = T T. Let e i be the basis (T i ) 0 i<p. Then we have (T i T j ) = (T i+j if i + j < p) and (0 otherwise) ɛ(t i ) = (1 if i = 0) and (0 otherwise) e(1) = T 0 m(t i ) = 0 j i i(t i ) = ( 1) i T i. ( ) i T i T i j j If {u i } is the dual basis of A. Then one can check that the -linear map A A sending u i to T i /i! is an isomorphism of Hopf algebras, and therefore (α p, ) α p,. We finish the talk with the following Proposition. Let k be any field with char k = p > 0. The the group schemes Z/pZ k, p,k and α p,k are pairwise non-isomorphic. Proof. The group scheme Z/pZ is isomorphic to Spec k p, so it is a reduced scheme, while p,k = Spec k[t ]/(T p k 1) and α p,k = Spec k[t ]/(T p ) are non-reduced, so they are distinct. As we have seen before p,k and α p,k are isomorphic as schemes. However as affine group schemes, they are not, for their Cartier duals are different from the previous two examples. 6

7 Bibliography [1] Finite Group Schemes, ichard Pink, CompleteNotes.pdf [2] Modular Forms and Fermat s Last Theorem, Chapter V, Finite Group Schemes 7