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 , 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 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 , 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 , Ri) F , 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
, 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)
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