56 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
R D , so if (1.4.4.2) is an isomorphism then we obtain a local W (κ)-algebra
homomorphism R D . Hence, R[1/p] = 0, so by [54, 7.1.9] (which interprets the
closed points of Spec(R[1/p]) in terms of rigid-analytic geometry) there is a local
W (κ)-homomorphism from R into the valuation ring of a field F of finite degree
over W (κ)[1/p]. The image of R in OF is a (possibly non-normal) local domain
that is finite flat over W (κ) and has residue field κ. (See 2.1.1 for another instance
of this slicing argument in deformation rings.)
To summarize, if (1.4.4.2) is an isomorphism then the problem of finding a
formal lift of X to characteristic 0 with residue field κ is reduced to the same
problem for with residue field κ . In other words, the formal flat lifting problem
with a fixed residue field is unaffected by making a preliminary scalar extension to
κ ! This is useful when κ = κ because it is often easier to make constructions
without increasing the residue field when the residue field is algebraically closed.
The problem of compatibility of the deformation ring with respect to change of
the coefficients (i.e., the isomorphism problem for analogues of (1.4.4.2)) arises in
applications ranging from moduli problems in algebraic geometry to Galois defor-
mation theory and beyond. In the special case that Λ is the finite ´ etale extension
of Λ corresponding to a finite separable extension κ of κ, an affirmative answer
to the isomorphism problem for the pair (Λ, Λ ) can be established within the ax-
iomatic deformation theory framework presented by Rim in [100, 1.15–1.19], which
considers more general Λ for which κ is just finitely generated over κ. However,
to include the case κ = κ all finiteness hypotheses on κ must be avoided. There
is an axiomatic approach to the “change of coefficients” problem (in the spirit of
Schlessinger’s criteria), independent of the methods of SGA7, but for the present
purposes it is simpler to give direct proofs in the cases we need. Later we will need
the abstract criterion, at which point we will state and prove it. (See Proposition
1.4.5.6.)
For the formal deformation rings of abelian varieties and p-divisible groups
(without extra structure), the isomorphism property for the analogue of (1.4.4.2)
is easy to verify:
1.4.4.13. Proposition. Let Λ Λ be a local map between complete local noe-
therian rings, inducing an extension κ κ of residue fields. Let G0 be an abelian
variety or p-divisible group over κ, and G0 = (G0)κ . In the case of p-divisible
groups, assume char(κ) = p.
For the deformation rings R and R pro-representing DefΛ(G0) and DefΛ (G0)
respectively, the natural map
R Λ ⊗ΛR
is an isomorphism.
Proof. In both cases, the map in question is between formal power series rings
over Λ . Hence, it suffices to check that the induced map between relative tangent
spaces over κ is an isomorphism. In each case, the tangent map is identified with
the natural map
(1.4.4.3) κ ⊗κ (Lie(G0)
t
⊗κ Lie(G0))
Lie((G0)t)
⊗κ Lie(G0)
that is an isomorphism. Indeed, in the case of abelian varieties this compatibil-
ity follows from the functoriality in the ground field for the general identification
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