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Dr. Katz DEq Homework Solutions 23

# Dr. Katz DEq Homework Solutions 23 - Algebraic Solutions of...

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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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