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 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 = 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
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