64 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

fact F (Λ , R ) always consists of a single element when it is non-empty, even when

R is not noetherian. The key point is that

mn

= 0 for some n 1, so we can

apply:

1.4.5.7. Lemma. Let R be a ring in which a prime p is nilpotent and let J ⊂ R

be an ideal such that Jn = 0 with n 1. Let f : X → Y be a homomorphism

between p-divisible groups over R. If f vanishes modulo J then f = 0.

Proof. The case n = 1 is trivial, and by induction on n we may assume n = 2.

Our problem is comparing two R-homomorphisms with the same reduction mod-

ulo J. Equipping J with trivial divided powers, this problem will be solved used

Grothendieck–Messing theory. Consider the Lie algebras of the universal vec-

tor extensions E(X) and E(Y ), equipped with their respective Hodge subbundles

Lie(Xt)∨ and Lie(Y t)∨. For any R-homomorphism u : X → Y , Grothendieck–

Messing theory shows that the map Lie(E(u)) respecting Hodge subbundles uniquely

determines u and only depends on u0 := u mod J. Hence, if u0 = 0 then u = 0.

We now apply Proposition 1.4.5.6 to the functor F defined above whose value

on any (Λ , R ) ∈ InfΛ is empty or a singleton. The established part (1) of Theorem

1.4.5.5 (with varying Λ) implies the pro-representability hypothesis in Proposition

1.4.5.6. Thus, we just have to verify conditions (i) and (ii) in Proposition 1.4.5.6.

Condition (ii) is immediate via Proposition 1.4.4.11. To establish (i) for (Λ , R ) =

lim(Λ

− →

, Ri), we just have to show that if F (Λ , R ) is non-empty then so is F (Λ , Ri)

for some i. We may rename Λ as Λ, R as R, and Ri as Ri for simplicity of notation.

Our problem is to show that if f0 lifts to an R-homomorphism f : XR → YR then

f descends to an Ri-homomorphism XRi → YRi for some large i. We shall induct

on the integer n 1 such that the maximal ideal m of R and maximal ideal mi of

every Ri have vanishing nth power, the case n = 1 being trivial.

Let R = R/mn−1 and Ri = Ri/mi

n−1,

so lim

− →

Ri = R. We may assume n 2

and (by induction) that the lift f := fR of f0 descends to a (necessarily unique) lift

f

i0

: XRi0 → YRi0 of f0. For i i0 let f

i

= f

i0

⊗Ri0 Ri. For i i0, if there is a

lift fi : XRi → YRi of f

i

then fi ⊗Ri R is a lift of f0 and thus coincides with f (by

Lemma 1.4.5.7). Hence, it is necessary and suﬃcient to find i i0 so that f

i

lifts

over Ri.

By Grothendieck–Messing theory (see [75, IV, 2.5; V, 1.6]), for every i i0

there is a canonical map

Li : Lie(E(XRi )) → Lie(E(YRi ))

that only depends on f

i

and has reduction Lie(E(f

i

)) modulo the square-zero ideal

mi

n−1

⊂ Ri, and moreover Li respects the Hodge subbundles if and only if f

i

lifts to an Ri-homomorphism XRi → YRi . Thus, it is necessary and suﬃcient to

prove that Li respects the Hodge subbundles for large i. Compatibility with base

change ensures that Li = Li ⊗Ri Ri whenever i i i0 and that Li ⊗Ri R =

Lie(E(f)). But this latter map respects the Hodge subbundles since it arises from

an R-homomorphism f lifting f. Hence, by standard limit arguments (and the

compatibility of the Hodge subbundles with respect to base change) it follows that

Li respects the Hodge subbundles for suﬃciently large i. This completes the proof

of Theorem 1.4.5.5.