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
= 0 for some n 1, so we can
220.127.116.11. 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 18.104.22.168 to the functor F defined above whose value
on any (Λ , R ) ∈ InfΛ is empty or a singleton. The established part (1) of Theorem
22.214.171.124 (with varying Λ) implies the pro-representability hypothesis in Proposition
126.96.36.199. Thus, we just have to verify conditions (i) and (ii) in Proposition 188.8.131.52.
Condition (ii) is immediate via Proposition 184.108.40.206. To establish (i) for (Λ , R ) =
, 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
Ri = R. We may assume n 2
and (by induction) that the lift f := fR of f0 descends to a (necessarily unique) lift
: XRi0 → YRi0 of f0. For i i0 let f
⊗Ri0 Ri. For i i0, if there is a
lift fi : XRi → YRi of f
then fi ⊗Ri R is a lift of f0 and thus coincides with f (by
Lemma 220.127.116.11). Hence, it is necessary and suﬃcient to find i i0 so that f
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
and has reduction Lie(E(f
)) modulo the square-zero ideal
⊂ Ri, and moreover Li respects the Hodge subbundles if and only if f
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 18.104.22.168.