1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 45

The analogous faithfulness result for p-divisible groups is more subtle when

char(κ) = p, since it is false without a noetherian condition:

1.4.4.2. Example. Let R = Zp[ζp∞ ] be the (non-noetherian) valuation ring of the

p-power cyclotomic extension of Qp, and let {ζpn } a compatible system of primitive

p-power roots of unity in R. The R-homomorphism between p-divisible groups

Qp/Zp → μp∞ defined by p−n → ζpn is an isomorphism between the generic fibers

and induces the zero map between the special fibers.

Under a noetherian hypothesis, the preceding pathology cannot occur:

1.4.4.3. Proposition. Let (R, m) be a noetherian local ring with residue field κ

of characteristic p 0. The functor G Gκ from p-divisible groups over R to

p-divisible groups over κ is faithful.

Proof. The problem is to prove that if f : G → G is a homomorphism between

p-divisible groups over R and fκ = 0 then f = 0. For each n 1, the induced map

fn : G

[pn]

→

G[pn]

between finite flat R-group schemes is described by a matrix

over R upon choosing R-bases of the coordinate rings (as finite free R-modules).

Hence, by the Krull intersection theorem it suﬃces to prove the vanishing result

over

R/mN

for all N 1, so we may and do assume that R is an artinian local

ring. By the functoriality of the connected-´ etale sequence, it suﬃces to treat the

following separate cases: G and G are both connected, G and G are both ´etale,

or G is ´ etale and G is connected.

The case when both are ´ etale is obvious. When G is ´ etale and G is connected

then we claim that Hom(G , G) = 0. By faithfully flat base change to the (artinian

local) strict henselization

Rsh

we may assume G is constant, so it is a power of

Qp/Zp. Hence, we can assume G = Qp/Zp, so Hom(G , G) = lim

←−

G[pn](R)

(inverse

limit via p-power maps). The equivalence between connected p-divisible groups and

formal Lie groups over R on which multiplication by p is an isogeny (see Example

1.4.3.6) identifies this inverse limit with the p-adic Tate module of G(R), where G

is the formal Lie group associated to G. Hence, the desired vanishing is reduced to

proving that G(R) has no non-zero infinitely p-divisible elements. In fact, we claim

that [pN ] kills G(R) for suﬃciently large N.

Upon choosing formal parameters for G, we may identify the set G(R) with

the set of ordered d-tuples in m, where d = dim G. If g ∈ G(R) has coordinates in

an ideal I of R then [p](g) has coordinates in (pI, I2) since [p] has linear part given

by p-multiplication on the coordinates. Hence, if we define the sequence of ideals

J0 = m and Jn+1 = (pJn,Jn) 2 then we just need JN = 0 for suﬃciently large N.

More generally, for any ring whatsoever and any ideal J0, an elementary induction

argument shows that Jn ⊂ (p, J0)n. The nilpotence of m then does the job.

Finally, we address the most interesting case, which is connected G and G.

In this case we switch to the perspective of formal Lie groups and aim to prove

that for commutative formal Lie groups Γ and Γ over R such that [p]Γ is an

isogeny, Hom(Γ , Γ) → Hom(Γκ, Γκ) is injective. Consider f ∈ Hom(Γ, Γ ) that

vanishes modulo an ideal I ⊆ m. Choose formal coordinates {xi} and {xj} for Γ

and Γ respectively, so the coeﬃcients of all monomials in f

∗(xj)

lie in I. Hence,

the formal power series (f ◦ [p]Γ

)∗(xj)

= [p]Γ

∗

(f

∗(xj))

has all coeﬃcients of all