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.
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