50 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

Proof. Since G and E are p-power torsion sheaves, the sheaf Y is the union of

its subsheaves Y

[pn]

for n 1. The snake lemma gives that [p] : Y → Y is an

epimorphism and provides short exact sequences of abelian sheaves

0 →

G[pn]

→ Y

[pn]

→

E[pn]

→ 0

for all n 1. The outer terms are represented by finite flat R-group schemes, so

by Proposition 1.4.1.3 the middle term must also be represented by a finite flat

R-group scheme. Thus, Y is a p-divisible group.

Over any base scheme S, the study of extensions of an ´ etale p-divisible group E

by a given p-divisible group G over S can be reduced to the special case E = Qp/Zp

at the cost of replacing G with a p-divisible group denoted

E∨

⊗ G, as follows.

Let

E∨

⊗ G be the direct limit (over n → ∞) of the tensor products

E[pn]∨

⊗Z/(pn)

G[pn]

of

G[pn]

against the

Z/(pn)-linear

dual

E[pn]∨

of the ´ etale sheaf

E[pn],

using the

evident transition maps. This is easily seen to be a p-divisible group. Note that if

S is the spectrum of a complete local noetherian ring with residue characteristic p

and if G is connected then E∨ ⊗G is connected with dimension height(E)·dim(G).

There is a general categorical equivalence (compatible with base change) from

the category of extensions of E by G to the category of extensions of the constant

p-divisible group Qp/Zp by E∨ ⊗ G. In one direction, for an extension (1.4.4.1)

apply E∨ ⊗ (·) and then pull back the short exact sequence

0 →

E∨

⊗ G →

E∨

⊗ Y →

E∨

⊗ E → 0

along Qp/Zp →

E∨

⊗ E corresponding to the identity map in

E[pn]∨

⊗

E[pn]

=

End(E[pn])

for n 1. In the other direction, given an extension of Qp/Zp by

E∨

⊗ G, we apply E ⊗ (·) to the given exact sequence and push out along the

evaluation map E ⊗

E∨

⊗ G → G. In an evident way, these are quasi-inverse

constructions.

To summarize, the formal smoothness of DefΛ(X0) for disconnected X0 is re-

duced to cases with X0 ´ et = Qp/Zp. Since the deformation theory is formally smooth

in the connected case, the formal smoothness of DefΛ(X0) is reduced to the follow-

ing assertion.

1.4.4.9. Lemma. Let (R, m) be artinian local with residue field κ of characteristic

p 0, J a non-zero ideal in R such that mJ = 0, and R0 := R/J. For a connected

p-divisible group G over R with reduction G0 over R0,

ExtR(Qp/Zp,G)

1

→ ExtR0

1

(Qp/Zp,G0)

is surjective.