1.4.
DIEUDONN´
E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 51
Proof. As above, let CR denote the category of finite R-algebras, equipped with
the fppf topology, and define CR0 similarly; the Ext-groups are computed in the
categories of abelian fppf sheaves on the respective sites CR and CR0 . By writing
Q/Z as the direct sum of its p-primary part Qp/Zp and its prime-to-p part M,
ExtR(Q/Z,G)
1
= ExtR(Qp/Zp,G)
1
ExtR(M,
1
G).
The final Ext-term vanishes: any extension E of M by G is a torsion sheaf, so the
decomposition of E as a direct sum of its p-primary part and prime-to-p part splits
the extension structure. Thus, our problem is proving the surjectivity of
ExtR(Q/Z,G)
1
ExtR0
1
(Q/Z,G0).
Consider the evident commutative diagram of long exact sequences
0 G(R) ExtR(Q/Z,G)
1
f
ExtR(Q,G)
1
f
H1(R,
G)
f
0 G0(R0) ExtR0
1
(Q/Z,G0) ExtR0
1
(Q,G0)
H1(R0,G0)
We want f to be surjective. The functor G on CR is pro-represented by the coordi-
nate ring O(G) of the associated formal Lie group, so it is formally smooth. Hence,
the left vertical map is surjective, so by the 5-lemma it suffices to show that f is
surjective and f is injective.
We first show that f is injective, for which it suffices to show that any G-torsor
fppf sheaf of sets X on CR is formally smooth. There is a local finite flat cover R of
R such that X|CR is pro-represented by O(G)R , so X is pro-represented by an R-
descent A of O(G)R ; this descent is easily checked to be a complete noetherian local
R-algebra (and its functor on CR is computed using local ring homomorphisms).
Clearly A is R-flat and the scalar extension κ
⊗κ over the residue field κ of
R is a formal power series ring over κ . It follows that k ⊗κ is regular for
any finite extension k of κ (it suffices to consider k containing κ , as a noetherian
ring with a regular faithfully flat extension is regular [73, Thm. 23.7]), so is
“geometrically regular” over κ. Thus, is formally smooth over κ relative to its
max-adic topology [73, Thm. 28.7], so R-flatness ensures that A is formally smooth
over R relative to its max-adic topology by [34, 0IV, 19.7.1].
It remains to show that f is surjective, so choose a short exact sequence
0 G0 E0 Q 0
representing a class ξ0 ExtR0
1
(Q,G0). The vanishing of
H1(S0,G0)
for all finite
R0-algebras S0 implies that we obtain a short exact sequence on S0-points for any
S0, and constant Zariski sheaves on CR are sheaves for the finite flat topology, so
pushforward along j : Spec(R0) Spec(R) gives an exact sequence
0 j∗(G0) j∗(E0) Q 0.
Since mJ = 0 and G is formally smooth, we have a short exact sequence
0 K G j∗(G0) 0
on CR where K (S) = Lie(Gκ) ⊗κ JS for any finite R-algebra S. Thus, p : G G
factors uniquely through a map h : j∗(G0) G since pK = 0, and the pushout of
j∗(E0) along h is an extension of Q by G whose pullback over R0 is pξ0. Since p
acts invertibly on Q, we conclude that f is surjective.
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