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October 17, 2014 P-DIVISIBLE GROUPS Let's Set Some Conventions. Let S = Spec R Where R Is a Complete Local Noetherian Ring. Le

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October 17, 2014 P-DIVISIBLE GROUPS Let's Set Some Conventions. Let S = Spec R Where R Is a Complete Local Noetherian Ring. Le October 17, 2014 p-DIVISIBLE GROUPS SPEAKER: JOSEPH STAHL Let's set some conventions. Let S = Spec R where R is a complete local noetherian ring. Let K = Frac R and k be the residue field of R. Recall that S is connected. 1. Reminder on finite flat group schemes Last time, we defined a group scheme over S to be a group object in S − Sch. If G is a finite flat group scheme then we have a connected-´etaleexact sequence j 0 ! G0 −!i G −! G´et ! 0: If we write G = Spec A then G0 = Spec A0 where A0 is the local quotient through which the co-unit of the Hopf algebra structure " : A ! R factors. And G´et = Spec(A´et) where A´et ⊂ A is the maximal ´etale subalgebra of A. Proposition 1. The two functors G 7! G0 and G 7! G´et are exact. Before we do this, let's recall something about ´etaleschemes. Let α : Spec k ! S be a geometric point. If X is an S-scheme then let X(α) = HomSch =S(Spec k; X) sep so if X = Spec A then X(α) = HomR(A; k). Let π1(S; α) = Gal(k =k). Then π1(S; α) acts on X(α) and X 7! X(α) defines an equivalence of categories between finite ´etale S-schemes and finite continuous π1(S; α)-sets. Proof. Let 0 ! G0 ! G ! G00 ! 0 be a short exact sequence in the category of group schemes over S. The easy directions are showing \connected parts" is left exact and \´etaleparts" is right exact. Let's do the ´etaleparts. Suppose that X is a finite flat group scheme over S such that the composition G0 ´et −! G´et −! X is zero. We know that we can make a larger commutative diagram containing G0 and G: G0 ~ " G G0 ´et 0 | G´et / X The composition G0 −! G −! G´et −! X is 0 and thus factors through the cokernel of G0 −! G. We also see that G −! G´et −! X must land in the ´etalepart of X, and hence the map factoring through G00 will factor further through G00 ´et. Hence, we get a unique map G00 ´et −! X, so G00 ´et satisfies the universal property of cokernels. Let's do the left exactness. Let α be a geometric point. Then the α points of the sequence are 0 ! G0(α) ! G(α) ! G00(α) is exact. Since G(α) = G´et(α) we get 0 ! G0 ´et(α) ! G´et(α) ! G00 ´et(α) is exact. Passing back through the equivalence of categories we are done. You can then show that the connected part is exact as well, by making an argument using orders. If 0 ! A ! B ! C is exact sequence of finite flat group schemes such that the order is multiplicative, then the sequence is exact on the right. 1 2. p-divisible groups We are now ready to talk about p-divisible groups proper. Let p be a prime. Definition. Let h ≥ 0.A p-divisible group over R of height h is a system (Gpn ; ιn) such that nh i) Gpn is a (commutative) finite flat group scheme over R of order p . n ιn p ii) The sequence 0 ! Gpn −! Gpn+1 −! Gpn+1 is exact. Before we think about this in the context of finite flat group schemes, think about it for finite abelian groups (i.e. what if an abelian group satisfies the previous properties). We get n h Gpn = (Z=p Z) : Q `i P n Indeed, Gpn = Z=p Z where `i = nh. But multiplication by p kills Gpn and so `i ≤ n for each i. Now its easy to see `i = n. G H Definition. Let G = (Gpn ; ιn ), H = (Hpn ; ιn ) be two p-divisible groups over S. A homomorphism G ! H is a family of group scheme morphisms fn : Gpn ! Hpn compatible with the ιn in each family; i.e., the diagram G G ι1 ι2 Gp / Gp2 / ··· f1 f2 H H ι1 ι2 Hp / Hp2 / ··· commutes and each fn is a morphism of finite flat group schemes over S. pn Example 2. Let A over S be an abelian scheme of dimension d. Then Apn = ker(A −! A). The ι are the natural inclusions. Then (Apn ; ιn) is a p-divisible group of height 2d. k p k Example 3. Gm(p) = (µpk ) where µpk = ker(Gm −! Gm). The order of µpk is p so Gm(p) is a p-divisible group of height 1. pk Iterating the ιn we get a map Gpk ! Gpk+` for any ` > 0. And they identity Gpk = ker(Gpk+` −! Gpk+` ). Then we can build the following exact sequence 0 ! Gpk ! Gpk+` ! Gp` ! 0: We start by considering the left exact sequence generated by the ι iteration: k ι ·p 0 / Gpk / Gpk+` / Gpk+` : We can extend this diagram to get G k+` ; pO 0 ·p` k ι ·p 0 / G k / G k+` / G k+` p p pO Gp` k n p Since Gpn is killed by multiplication by p , we have that Gpk+` −! Gpk+` factors uniquely through the kernel ` of multiplication by p , Gp` . We obtain the exact sequence 0 ! Gpk ! Gpk+` ! Gp` ; and at this point one can simply check orders to see that this is in fact right exact if we add a zero to the right. 2 Just like for finite flat group schemes, we have connected and ´etale p-divisible groups associated to a given p-divisible group, and moreover we have a connected-´etalesequence for p-divisible groups as well. 0 0 0 ´et ´et ´et Proposition 4. If G is a p-divisible group then ιn induce maps ιn : Gpn ! Gpn and ιn : Gpn ! Gpn and 0 0 ´et ´et (Gpn ; ιn) and (Gpn ; ιn ) are p-divisible groups. Proof. The exactness of the functors we previously mentioned implies the exactness of the defining exact sequence for p-divisible groups. It can be shown that both G0 and G´et have the proper orders at each finite level, so in particular, they have a height and hence are p-divisible groups in their own right. It makes sense now to define connected and ´etale p-divisible groups. Definition. A connected p-divisible group is a G such that G = G0. An ´etale p-divisible group is a G such that G = G´et. Note that a connected p-divisible group is one such that each Gpn is connected, and similarly for ´etale p-divisible groups. 3. A theorem Recall: for an elliptic curve E=K we define the `-adic Tate module T (E) = lim E[`n](K). If ` 6= char(K) ` − 2 then T`(E) ' Z` but if ` = char(K) then T`(E) is either zero or Z`. Theorem 5. Let E1 and E2 be elliptic curves over K and let ` 6= char(K) be a prime. Then Hom(E1;E2) ⊗ Z` ! HomGal(K=K)(T`(E1);T`(E2)) is injective. Unfortunately, this fails for ` = char(K). We will try to remedy this by using p-divisible groups. From a p-divisible group over R we can construct a p-divisible group over the residue field and fraction field. If G = (Gpn ) then GK = (Gpn ×R K) is the generic fiber and Gk = (Gpn ×R k). Theorem 6. If R is a complete noetherian local ring with residue characteristic p then (a) Homp−div=R(G; H) ! Homp−div=k(Gk;Hk) is injective. (b) If R is integrally closed then Homp−div=R(G; H) ! Homp−div=K (GK ;HK ) is a bijection. Remark. To a p-divisible group G, we may associate a Tate module T (G) = lim(G n ) (K). Then the second − p K part of the above theorem implies that for two p-divisible groups over R, Hom(G; H) ,! HomGK (T (G);T (H)) is a bijection, where GK is the absolute Galois group of K. We can illustrate that the theorem is true when p-divisible groups are replaced by elliptic curves. Let 0 0 0 E; E be elliptic curves over R and EK ;EK generic fibers and Ek and Ek the special fibers. 0 0 Let f : E ! E be a map over R. Then we see f = 0 if and only if fK : EK ! EK is zero. To prove injectivity of (1), suppose that fk = 0. By the previous point it is enough to show that fK = 0. The theorem we stated before with Tate modules says that we can check this after passing through the map 0 n 0 T`(EK ) ! T`(EK ) is zero, i.e. fK on EK [` ](K) is zero. But since EK and EK have good reduction, if m - char(k) we get EK [m](K) ,! Ek(k) is injective. However, while the theorem holds for elliptic curves, it fails for even simple examples of finite flat group schemes. For example, if we let R = Zp[ζp] and consider the map Z=pZ −! µp of group schemes given by 1 7! ζp, we know that in general the map will be nontrivial on points, essentially because R contains a pth root of unity. However, reduction mod p will yield algebras over Fp, which will not contain nontrivial pth roots of unity, so that any map to µp will be zero. In the ´etalecase we can provide a sketch of the theorem. 3 Proof. Begin with a map of ´etale p-divisible groups over R f : G −! H, and fix a geometric point α : Spec K −! Spec R.
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