52 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

We have settled the formal smoothness in Theorem 1.4.4.7. Before we address

the other assertions, we make a practical observation:

1.4.4.10. Remark. The infinitesimal liftability of p-divisible groups, Zariski lo-

cally on the base, is a hypothesis which underlies the Grothendieck–Messing crys-

talline Dieudonn´ e theory [75, Ch. IV, V] that classifies the deformations of p-

divisible groups. It is a deep theorem of Grothendieck–Illusie that this liftability

property holds without Zariski-localization over any aﬃne base on which p is nilpo-

tent (see [51, 4.4]). To apply Grothendieck–Messing theory over artinian local

base rings (which is all we really need in this book) it suﬃces to use the formal

smoothness established above.

Continuing with the proof of Theorem 1.4.4.7, we address Schlessinger’s cri-

teria for pro-representability of F = DefΛ(X0) (and compute its tangent space).

Consider a pair of maps R1,R2 ⇒ R0 in CΛ with R1 → R0 surjective. Since

deformations of p-divisible groups over artinian local rings have no non-trivial au-

tomorphisms, the bijectivity of

F (R1 ×R0 R2) → F (R1) ×F

(R0)

F (R2)

is immediate from part (2) of the following general result (which will be very useful

in our later work with algebraization of formal CM abelian schemes; see Theorem

2.2.3).

1.4.4.11. Proposition (Ferrand). Let p1 : R1 → R0 and p2 : R2 → R0 be maps

of rings with p1 surjective. Let R denote the fiber product ring R1 ×R0 R2.

(1) If M is a flat R-module and Mj denotes the flat Rj-module M ⊗R Rj then the

natural map M → M1 ×M0 M2 is an isomorphism. Conversely, if Mj is an

Rj-module and there are given isomorphisms R0 ⊗R1 M1 M0 R0 ⊗R2 M2

then for the R-module M = M1 ×M0 M2 the natural maps Rj ⊗R M → Mj

are isomorphisms, and M is R-flat when each Mj is Rj-flat.

(2) Let Rj be a finitely generated Rj-algebra and suppose there are given iso-

morphisms of R0-algebras R0 ⊗R1 R1 R0 R0 ⊗R2 R2. The R-algebra

R := R1 ×R0 R2 is finitely generated, and if each Rj is flat and finitely pre-

sented over Rj then R is flat and finitely presented over R.

(3) Assume that p2 is surjective or that all elements of ker(p1) are nilpotent. The

functor

X (XR1 , XR2 , (XR1 )R0 (XR2 )R0 )

from the category of flat R-schemes to the category of triples (X1,X2,f) con-

sisting of flat schemes Xj over Rj and an R0-isomorphism f : (X1)R0

(X2)R0 is an equivalence, and X is finite type (respectively flat and finitely

presented) over R if and only if each Xj is finite type (respectively flat and

finitely presented) over Rj.

An R-map f : X → Y between flat finitely presented R-schemes satisfies

property P if and only if the pullback maps fR1 and fR2 satisfy P, where P

is any of the properties: separated, proper, finite, flat, smooth, ´ etale, isomor-

phism, geometric fibers of pure dimension d, connected geometric fibers.

Generalizations of parts (2) and (3) are given in [101, Appendix A] (and ref-

erences therein). For applications to Schlessinger’s criteria, part (3) is used with