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 Aκ over the residue field κ of

R is a formal power series ring over κ . It follows that k ⊗κ Aκ 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 Aκ is

“geometrically regular” over κ. Thus, Aκ 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.