Algebraic Solutions of Differential Equations
23
(over Ao) divisors Di, o in X o which cross normally relative to So =
Spec(Ao), such that the geometric situation (1.0) over S (for the purposes
of 2.3.2, the base scheme T figuring in (1.0) may be taken to be S itself)
comes from the analogous situation over
So
by the change of base
S, So
(cf. EGA IV, 8.9.1, 8.10.5, and 7.7.9). We must show that after replacing
Ao by a larger subring At, A =At ~ Ao, which is still finitely generated
over Z, the hypotheses (2.3.2.1.2), that the Hodge ==De Rham spectral
sequence degenerate at E1 and have Et locally free of finite rank, are
valid over At, which is a noetherian ring. (For then, once the theorem is
proved over At, it remains true over A by (2.2.1.11).) In fact, it suffices to
find such an A1 over which (2.3.2.1) holds; then (2.3.2.2) follows. For if
we suppose that over At the E~ terms noted
E'~'b(AO,
are
locally free of
finite rank,
the differential
(2.3.2.4.1)
d t
E~'b(A~) ~
E~ + L b(A~)
must vanish, because after extension of scalars to A ~ Ax~ this differential
becomes zero, because of the commutative diagram (cf. (2.2.1.7))
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 Fall '11
 NormanKatz
 Ring, Abelian group, Free module, Module theory

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