62 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

Consider the set-valued functor F on CR that carries an object R to the set of

all R-homomorphisms XR → YR lifting f0 (i.e., F (R) is empty if there is no such

lift and F (R) has a single element when a lift exists). The problem is to show that

F is pro-represented by a quotient of R. By the functorial aspect of Proposition

1.4.4.11, it is straightforward to check that for a pair of maps R1,R2 ⇒ R0 in CR

at least one of which is surjective, the natural map

F (R1 ×R0 R2) → F (R1) ×F

(R0)

F (R2)

is bijective. Moreover, F (κ[ ]) consists of a single element (the constant deformation

of f0 as a homomorphism from X ⊗R κ[ ] = X0 ⊗κ κ[ ] to Y0 ⊗κ κ[ ]), so it vanishes

as a κ-vector space. Thus, by Schlessinger’s criteria, F is pro-represented by a

complete local noetherian R-algebra with residue field κ and has vanishing relative

tangent space (over R), so the ring pro-representing F is a quotient of R. This

completes the proof of (1).

In view of the proof of (1), to prove (2) it suﬃces to show that for (R,I) as

above and any local homomorphism R → R to a complete local noetherian ring

with residue field κ ⊃ κ, IR is the analogous ideal inside R relative to XR , YR ,

and (f0)κ . This seems diﬃcult to verify directly, so we digress to prove an abstract

isomorphism criterion for maps such as (1.4.5.2) and then apply it to establish (2).

To formulate an abstract isomorphism criterion for the “change of coeﬃcients”

map for deformation rings, we need to assume that the functor is defined on a

larger class of rings than the artinian ones. For a complete local noetherian ring Λ

with residue field κ, define InfΛ to be the category of pairs (Λ , R ) consisting of a

complete local noetherian Λ-algebra Λ and a local Λ -algebra (R , m) such that

(i) Λ → R is local and induces an isomorphism on residue fields,

(ii) mn = 0 for some n 1.

A morphism (Λ1,R1) → (Λ2,R2) consists of a local Λ-algebra map f : Λ1 → Λ2 and

a local homomorphism R1 → R2 over f. In an evident way, InfΛ contains CΛ as a

full subcategory for any Λ . Also, if n 1 and {(Λ , Ri)} is a directed system in InfΛ

such that mRi

n

= 0 for all i then R := lim

− →

Ri equipped with its evident Λ -algebra

structure is an object in InfΛ whose maximal ideal has vanishing nth power. As an

important special case, for any (Λ , R ) in InfΛ with mR n = 0, the directed system

{Ri} of artinian local Λ -subalgebras of R provides a directed system {(Λ , Ri)} in

InfΛ with all maximal ideals having vanishing nth power and lim

− →

Ri = R .

Consider a covariant set-valued functor F on InfΛ and any directed system

{(Λ , Ri)} as above. There is a natural map

lim

− →

F (Λ , Ri) → F (Λ , lim

− →

Ri).

If this is always bijective then we say that F commutes with direct limits. If we

only consider such directed systems with a fixed Λ (such as Λ) then we say F

commutes with direct limits over Λ . In each of these definitions it suﬃces to

consider direct limits with Ri that are artinian. For example, if (R, m) is a complete

local noetherian Λ-algebra with residue field κ and F is defined to be the functor

(Λ , R ) HomΛ(R,R ) (using local Λ-algebra maps) then F commutes with direct

limits because

R/mn

is artinian with finite Λ-length for every n.

Choose Λ with residue field κ and assume the restriction F |CΛ is pro-represent-

ed by a complete local noetherian Λ -algebra (R , m ) with residue field κ . Also

assume F |CΛ is pro-represented by a complete local noetherian Λ-algebra (R, m)