1.4.

DIEUDONN´

E THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 63

with residue field κ. For each n 1 there is a universal element ξn ∈ F

(R/mn),

so for any m 1 the induced element in F ((Λ /mΛ

m

) ⊗Λ

(R/mn))

is classified by a

map of local Λ -algebras

R → (Λ /mΛ

m

) ⊗Λ

(R/mn)

that is compatible with change in m and n (since the ξn’s are compatible with

change in n). Passing to the inverse limit defines a map of complete local noetherian

Λ -algebras

(1.4.5.3) R → Λ ⊗ΛR

that recovers (1.4.4.2) in the setting of Example 1.4.4.12.

We seek abstract conditions on F which ensure that the map (1.4.5.3) is an

isomorphism. Such an isomorphism property for all Λ (assuming F |CΛ is pro-

represented by a complete local noetherian Λ -algebra for all Λ ) says exactly that

F = HomΛ(R, ·), since we have seen the necessity of commutation with direct limits

in such cases and every object (Λ , R ) in InfΛ is the direct limit of its artinian local

Λ -subalgebras.

1.4.5.6. Proposition. Let Λ be a complete local noetherian Λ-algebra with residue

field κ , and let F be a covariant set-valued functor on InfΛ such that F |CΛ and

F |CΛ are pro-represented by complete local noetherian rings (R, m) and (R , m )

with residue fields κ and κ respectively. The map (1.4.5.3) is an isomorphism if

and only if the following conditions hold:

(i) F commutes with direct limits over Λ,

(ii) for any (Λ , R ) ∈ CΛ and the local Λ-subalgebra R = R ×κ κ ⊂ R with

residue field κ, the natural map F (Λ,R) → F (Λ , R ) is bijective.

This result is an abstract version of an argument of Faltings in the setting of

Galois deformations; see [129, pp. 457-8]. Note that R in (ii) is not noetherian

when [κ : κ] is not finite and R = κ .

Proof. The necessity of (i) has been explained, and the necessity of (ii) is obvious.

To prove suﬃciency, we first make a general construction that has nothing to do

with (i) or (ii).

For any (Λ , R ) in InfΛ, each local Λ-algebra map f : R → R factors through

a local Λ-algebra map fn :

R/mn

→ R for some n 1. The map

F (fn) : HomΛ(R,

R/mn)

= F

(R/mn)

→ F (R )

produces an element of F (R ) that is independent of n and functorial in (Λ , R ),

so it defines a natural transformation of functors HomΛ(R, ·) → F on InfΛ.

Our problem is precisely to prove that this is an isomorphism on CΛ . By (ii),

it suﬃces to work on the category of pairs (Λ,R). Now using (i), we are done.

The abstract criteria in Proposition 1.4.5.6 will now be used to establish part (2)

of Theorem 1.4.5.5 (taking Λ in the abstract criteria to be the universal deformation

ring of X0 as in Theorem 1.4.5.5). For a pair of p-divisible groups X and Y over Λ

and a homomorphism f0 : X0 → Y0 between the special fibers, define the covariant

set-valued functor F on InfΛ to carry (Λ , R ) to the set of deformations of (f0)κ to

R (where κ is the residue field of Λ ). The set F (Λ , R ) is empty when there is no

lift and it consists of a single element when there exists a lift and R is artinian (as

the uniqueness of such a lift for artinian R follows from Proposition 1.4.4.3). In