Dr. Katz DEq Homework Solutions 25

Dr. Katz DEq Homework Solutions 25 - and by (2.3.1.2) the...

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Algebraic Solutions of Differential Equations 25 while the first term is, by (2.3.1.2.1), (2.3.2.13) ~ lngA(co,E~'b) = ~ lngA (F~s Raf,(f2bx/s(log D))). a,b a,b Now by hypothesis (2.3.2.1), each of the A-modules Raf,(f2bx~s(log S)) is free of finite rank, and hence each of the A-modules F~*s Raf, (~/s(log D)) is free of the same finite rank. In particular, we have, for each a, b (2.3.2.14) lngA (F*s Raf, (f2bx/s (log D))) = lnga (Raf, (f2bX/S (log D))). Putting together (2.3.2.12-14), the criterion (2.3.2.11) for degeneration at E2 may be written (2.3.2.15) ~ lnga(Raf,(f2bx/s(log D)))=~ lngA(R"f,(f2~/s(log D))). a,b n This last equality holds, in virtue of the hypothesis (2.3.2.2) that the Hodge => De Rham spectral sequence is degenerate at E~. This concludes the proof of degeneration. Q.E.D. (2.3.2.16) Corollary. Under the hypotheses of proposition (2.3.2), the formation of the conjugate spectral sequence (2.3.0.1) commutes with arbitrary change of base S'--* S. Proof. By (2.3.2), the conjugate spectral sequence is degenerate at E2,
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Unformatted text preview: and by (2.3.1.2) the formation of its E 2 term commutes with arbitrary change of S'~ S. The result follows by (2.2.1.11). (2.3.3) We are now in a position to explain the terminology "conjugate" spectral sequence. With the assumptions of proposition (2.3.2), suppose further that S is the spectrum of a field K of characteristic p, and that the divisor D is void. The Hodge ~ De Rham spectral sequence (2.3.3.1) E~' b = H b (X, ~X/K) =*" Ha + b (X, ~"~'/K) being degenerate at Et, we have (2.3.3.2) g~ (H n (X, f2~/r ) ) ~ H"- ~ (X, f~a'/~)- The degeneracy at E2 of the conjugate spectral sequence (2.3.3.3) co,E~2'b =F~H~(X, f2~,/~) =~ H~+~(X, f2~/r) which we prefer to rewrite as (2.3.3.4) r =H~(X ~p~, O~c~/K) =~ Ha+b(X, f2~//~), gives an isomorphism (2.3.3.5) g~on n~ (X, f2~c/K ) ~-- H a (X ~p~, f~}?~,~/K) Putting together (2.3.3.5) and (2.3.3.3) (for X tp~) we find an isomorphism (2.3.3.6) g~o H"(X, f2~/r)"~g~-~ H~(X ~v~, I2],~,/r ) ....
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