Download October 17, 2014 p-DIVISIBLE GROUPS Let`s set some conventions

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-étale exact sequence
j
i
0 → G0 −→ G −→ Gét → 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 ε : A → R factors. And Gét = Spec(Aét ) where Aét ⊂ A is the maximal étale
subalgebra of A.
Proposition 1. The two functors G 7→ G0 and G 7→ Gét are exact.
Before we do this, let’s recall something about étale schemes. Let α : Spec k → S be a geometric point.
If X is an S-scheme then let
X(α) = HomSch /S (Spec k, X)
so if X = Spec A then X(α) = HomR (A, k). Let π1 (S, α) = Gal(k sep /k). Then π1 (S, α) acts on X(α)
and X 7→ X(α) defines an equivalence of categories between finite étale 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 “étale parts” is right exact.
Let’s do the étale parts. Suppose that X is a finite flat group scheme over S such that the composition
G0 ét −→ Gét −→ X is zero. We know that we can make a larger commutative diagram containing G0 and
G:
G0
G
"
~
|
G0 ét
0
/X
Gét
The composition G0 −→ G −→ Gét −→ X is 0 and thus factors through the cokernel of G0 −→ G. We
also see that G −→ Gét −→ X must land in the étale part of X, and hence the map factoring through G00
will factor further through G00 ét . Hence, we get a unique map G00 ét −→ X, so G00 ét 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) = Gét (α) we get
0 → G0 ét (α) → Gét (α) → G00 ét (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.
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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
i) Gpn is a (commutative) finite flat group scheme over R of order pnh .
pn
ι
n
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
Gpn = (Z/pn Z)h .
Indeed, Gpn = Z/p`i Z where
its easy to see `i = n.
Q
P
`i = nh. But multiplication by pn kills Gpn and so `i ≤ n for each i. Now
H
Definition. Let G = (Gpn , ιG
n ), 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
Gp
ιG
1
/ Gp2
f1
Hp
ιH
1
ιG
2
/ ···
ιH
2
/ ···
f2
/ Hp 2
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.
pk
Example 3. Gm (p) = (µpk ) where µpk = ker(Gm −→ Gm ). The order of µpk is pk 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:
0
/ Gpk
ι
/ Gpk+`
·pk
/ Gpk+` .
We can extend this diagram to get
Gpk+`
; O
0
0
/ Gpk
ι
/ Gpk+`
·p`
·p
k
/ Gpk+`
O
Gp`
pk
Since Gpn is killed by multiplication by pn , 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.
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Just like for finite flat group schemes, we have connected and étale p-divisible groups associated to a given
p-divisible group, and moreover we have a connected-étale sequence for p-divisible groups as well.
ét
ét
Proposition 4. If G is a p-divisible group then ιn induce maps ι0n : G0pn → G0pn and ιét
n : Gpn → Gpn and
ét
0
ét
0
(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 Gét 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 étale p-divisible groups.
Definition. A connected p-divisible group is a G such that G = G0 . An étale p-divisible group is a G such
that G = Gét .
Note that a connected p-divisible group is one such that each Gpn is connected, and similarly for étale
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)
←−
then T` (E) ' Z2` 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(Gpn )K (K). Then the second
←−
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
E, E 0 be elliptic curves over R and EK , EK
generic fibers and Ek and Ek0 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
0
T` (EK ) → T` (EK
) is zero, i.e. fK on EK [`n ](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 étale case we can provide a sketch of the theorem.
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Proof. Begin with a map of étale p-divisible groups over R f : G −→ H, and fix a geometric point α :
Spec K −→ Spec R. Setting G(α) := lim Gpn (K), we see that G(α) is a free Zp -module, as each Gpn (K) ∼
=
←−
(Z/pn Z)h . Then apply the equivalence of categories between finite étale schemes and finite continuous
π1 (R, α) sets and take the inverse limit to obtain a map f : G(α) −→ H(α) of free Zp -modules with
continuous actions of π1 (R, α) (in fact, G 7→ G(α) gives an equivalence of categories between étale p-divisible
groups over R and free Zp -modules with continuous π1 (R, α) action). If Gk and Hk denote the p-divisible
groups over the special fiber, α0 : Spec k −→ Spec k is a fixed geometric point, and Gk (α0 ) = lim Gk,pn (k)
←−
and Hk (α0 ) = lim Hk,pn (k) are the free Zp -modules associated to Gk and Hk , we see that Gk (α0 ) = G(α) as
←−
modules (Gk,pn (k) is still just (Z/pn Z)h as a group, although we now have a continuous action of π1 (k, α0 )
rather than an action of π1 (R, α)). Let f k : Gk (α0 ) −→ Hk (α0 ) be the map corresponding to fk : Gk −→ Hk .
We know f k = 0 since fk = 0 by assumption, but the equality of modules G(α) = Gk (α0 ) implies that f = 0
as well, which in turn implies that f = 0, using the equivalence of categories described at the beginning of
the proof.
If G and H are connected rather than étale, we can analyze their relationship by examining formal Lie
groups, which is the topic of the next talk.
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