NOTES ON CARTIER DUALITY
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1 NOTES ON CARTIER DUALITY Let k be a commutative ring with, and let G = SpecA be a commutative finite locally free group scheme over k. We have the following k-linear maps unity i: k A, multiplication m: A k A A, inverse τ : A A, co-unity ε : A k, co-multiplication µ : A A k A satisfying the axioms for a commutative co-commutative Hopf algebra over k. Recall that A is a projective k-module of finite rank. Let A := Hom k A,k. Dualizing the structure maps for A, we get k-linear maps unity i : k A, multiplication m : A k A A, inverse τ : A A, co-unity ε : A k, co-multiplication µ : A A k A making A a commutative co-commutative Hopf algebra over k. Here i is the transpose of ε, m is the transpose of µ, τ is the transpose of τ, ε is the transpose of i and µ is the transpose of m. Let Ĝ = SpecA be the commutative finite locally free group scheme attached to A. For every commutative k-algebra R, we use a subscript R to denote the base-changed objects like A R := A k R, A R = A k R = Hom R A R,R and for morphisms like µ R : A R A R R A R, ε R : A R R. The set of R-valued points GR = Hom k-alg A,R Hom k R,A R = A R of G is identified with the set of all ˆφ A R satisfying the properties i and ii below. i µ R ˆφ = ˆφ ˆφ A R R A R ii ε R ˆφ = R. ii ˆφ A R. Note that i says that the transpose φ : A R R of ˆφ respects multiplication, while ii says that φ i R = id R. So i and ii says that φ is an homomorphism of k-algebras. On the other hand, the set Hom R-grp Ĝ Speck SpecR,G m SpecR of all R-homomorphisms from Ĝ Speck SpecR to G m SpecR is naturally identified with the set of all elements ˆφ A R satisfying conditions i and ii. Lemma. Suppose that ˆφ A R satisfies i. Then ii ii. In other words, GR is in natural bijection with H om k G, G m R. PROOF. ii = ii. Apply the identity = in A R to ˆφ, we get ε R ε R µ R = ε R ε R ˆφ 2 = ε R ˆφ Hence ε R ˆφ = because ε R ˆφ is a unit in A R by ii. ii = ii. Apply the identity for the additive inverse in Ĝ R := SpecA R m R A τ R µ R = i R ε R to ˆφ, we get ˆφ τ R ˆφ = in A R. So ˆφ is a unit in A R.
2 Apply the above Lemma to Ĝ, we see that sheaf H om k G, G m of commutative groups over Speck is representable, and naturally identified with Ĝ = SpecA as schemes at this point. One can reformulate this as a morphism can G : G Speck Ĝ G m Speck obtained from the above Lemma applied to the tautological element in GA with R = A. This morphism corresponds to the k-algebra homomorphism k[t,t ] A k A which sends T to the diagonal element δ A k A which corresponds to Id A. Since δ also corresponds to Id A, the canonical morphism can G : is naturally identified with canĝ. Moreover the Lemma tells us that µ Aδ = δ A δ A k A k A, ε Aδ = i A, and δ τ Aδ =. The same argument because δ also correspond to the tautological element in ĜA gives µ A δ = δ A δ A k A k A, ε A δ = i A and δ τ A δ =. Note that δ A δ is the product of pr 2 δ and pr 3 δ in A k A k A, and δ A δ is the product of pr 3 δ and pr 23 δ in A k A k A. The formulas and gives the multiplicative inverse of δ in A k A, namely τ A δ = τ A δ. More importantly they also show that the canonical map can: G Speck Ĝ G m Speck is bi-multiplicative. Example. Let H be an abstract commutative finite group. Write k H for the set of all k-valued functions on H, and k[h] be the group algebra of H over k. The delta functions δ h at h H form a k-basis of k H, and we have δ x δ y = δx,yδ x for all x,y H, where δx,y denotes the Kronecker s symbol. The co-multiplication, co-unit and inverse in k H are given by µ : δ h δ x δ y, ε : δ x δx,0δ x, τ : δ x δ x. x,y H, x y=h The group algebra k[h] is best thought of as the convolution algebra of all k-valued measures on H, where the basis element [h] corresponding to an element h H is evaluation at h. The comultiplication, co-unit and inverse are given by µ : [x] [x] [x], ε : [x], τ : [x] [ x]. Some samples of the equalities in and are: µ k[h] δ x [x] = δ y δ z [y + z] = y,z H in k H k k H k k[h], ε k H δ x [x] = δ x = i k H in k H, and δ x [x] δ y [y] y H = δx,yδ x [x y] = x,y H δ y [y] y H δ z [z] z H δ x [0] = i k H k[h] x in k H k k[h]. When H = Z/nZ we have Speck[Z/nZ] = Speck[T ]/T n = µ n Speck. 2
3 Example 2. Let p be a prime number, k F p be a field, G = α p Speck = Speck[X]/X p. Let x A = k[x]/x p be the image of X in A. The co-multiplication and co-unity are determined by µ : x x + x and ε : x 0. Let y 0,y,...,y p A = Hom k A,k be the dual basis of,x,x 2,...,x p. Then we have µ : y i y a y i a, y i = i!y i i = 0,,..., p, y p = 0. 0 a i Then x y establishes an isomorphism A = A of Hopf-algebras. The diagonal element δ = p x i y i A k A i=0 is equal to expx y = + x y + x2 y 2 2! where E p T is the truncated exponential + + xp y p p! = E p xy, E p T := + T + T 2 2! + + T p p! k[t ] In other words, the formula x,y Exy gives an auto-duality pairing which identifies α p with its own Cartier dual. α a α p G m Example 2. Let k F p be a field of characteristic p. Let Fr p n : G m Speck G m Speck be the Frobenius homomorphism defined by X X pn. Denote by α p n = Speck[X]/X pn the kernel of Fr p n. We have short exact sequences 0 α p n j n,n+m β n+m,m α p n+m α p m 0 for positive integers m,n, where j n,n+m is the natural inclusion and β n+m,m is induced by Fr p n. Write A := k[x]/x pn and let x be the image of X in A. Let y 0,y,...,y p n be the k-basis in A := Hom k A,k dual to the k-basis,x,x 2,...,x p of A. The co-multiplication on A is given by µ : x x + x. The co-multiplication, unity and co-unity on A are given by µ : y i y a y i a i = 0,,..., p n ; i : y 0, ε : y i 0 i > 0. 0 a i It is straight forward to deduce from µx = x + x that y 2 = 2y 2, y 3 = 3!y 3,...,y p = p! y p, y p = 0. 3
4 Similarly we have y j p a = j! y jp a and yp pa = 0 a = 0,,...,n, j = 0,,..., p. More generally, for every natural number i with 0 i p n, written in p-adic expansion in the form i = 0 a j a p a, then y j a p a y i = y j0 + j p+ + j p = j a!. 0 a So A is isomorphic to k[z 0,Z,...,Z ]/Z p 0,Zp,Zp 2,...,Zp as k-algebras, such that y p a corresponds to the image of Z a for a = 0,,...,n. The diagonal element δ = p n x i y i A k A = k[x,z0,z,...,z ]/X pn,z p 0,Zp,...,Zp i=0 can be written in terms of the truncated exponential E p T = + T + T 2 /2! + + T p /p! as the image of the polynomial δx,z = δx,z 0,Z,...,Z = a=0 E p X pa Z a in k[x,z 0,Z,...,Z ]/X pn,z p 0,Zp,...,Zp. The group law of the Cartier dual α p of α p n is completely determined by the polynomial δx,z as follows. Using Z 0,Z,...Z as the coordinates on α p, then the the sum of two points in α p n with coordinates z = z 0,z,...,z, w = w 0,w,...,w is the point with coordinates Φz,w where ΦZ, W = Φ 0 Z, W,...,Φ Z,W n k[z 0,Z,...,Z,W 0,W,...,W ]/Z p 0,...,Zp p,w p 0,...,W p is determined by the equation E p X pa Φ a Z,W a=0 = a= E p X pa Z a a= E p X pa W a in k[x,z 0,Z,...,Z,W 0,W,...,W ]/X pn,z p 0,...,Zp p,w p 0,...,W p. Notice that but E p XZ + XW E p XZ E p XW mod X p,z p,w p, E p XZ 0 +W 0 +X p Z +W E p XZ 0 +X p Z E p XW 0 +X p W mod X p2,z p 0,Zp,W p 0,W p. So the usual exponential rule does not hold for the truncated exponential when applied to rings like k[x,z 0,Z,W 0,W ]/X p2,z p 0,Zp,W p 0,W p. The Cartier dual of the homomorphism β n+m,m : α p n+m α p m power of the Frobenius, corresponds to the homomorphism induced by Fr p n : x x pn, the n-th β n+m,m : k[y 0,...,Y n+m ]/Y p n+m k[y 0,...,Y m ]/Y p m 4
5 of Hopf algebras such that β n+m,m : Y 0,...,Y 0; β n+m,m : Y n+a Y a, a = 0,...m. Similarly the natural embedding j n,n+m : α p n α p n+m corresponds to the homomorphism j n,n+m : k[y 0,...,Y ]/Y p k[y 0,...,Y n+m ]/Y p n+m of Hopf algebras which sends each Y a to Y a for all a = 0,,...,. Using the maps j m,m 2 it is easy to see that for each natural number a with 0 a n, the a-th component Φ a Z, W of the group law comes from a unique polynomial in F p [Z 0,...,Z a,w 0,...,W a ] independent of n whose degree in each variable is p. For instance p Z0 Φ 0 Z, W = Z 0 +W 0, Φ Z, W = Z +W + i i= i! W p i 0 p i!. 5
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