Dr. Katz DEq Homework Solutions 57

Dr. Katz DEq Homework Solutions 57 - (4.0.1.5)...

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Algebraic Solutions of Differential Equations 57 The "Poincar6 residue" (cf. [7, 8]) furnishes isomorphisms (which are ,, canonical" only after choosing an ordering of the finite set which indexes the smooth divisors Di) I O if n<O (4.0.1.2) grW(f2}/s(logD~= J c~ f2.-, . IJ / il<ff<in Dil ..... Oln/S if n> 1 t t?}/s if n = 0. [We are guilty of an abuse of notation in (4.0.1.2) above, because, strictly speaking, gr,, w is isomorphic to the direct sum of the extensions by zero from Diln. ..~Di. to X of the complexes f2g~,~" .... oi./s]. To avoid either further abuses or typographical disaster, we will henceforth adopt the following notation: eat f(i 1 .... , i,) denote the composite Diln. ..nDi c--~Dc--, X (4.0.1.3) s. IS S and let us abbreviate, for each integer k > 0, (4.0.1.4) HkR(DilA. ..ADi./S) d~ n Rkf(il, . ..,in),(~Diln" .... Din/S) , which is a quasicoherent sheaf of d?s-modules. The spectral sequence of the filtration W and the functor R f, may be written, via (4.0.1.2), as
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Unformatted text preview: (4.0.1.5) wE;&quot;'k+&quot;= Hk-R&quot;(Di, n'&quot;nDi./S)~ Rkf.(t2;x./s(logD)). il &lt;'&quot;&lt;in (4.0.2) Proposition. The spectral sequence (4.0.1.5) is (naturally) a spectral sequence with cup-product in the category of quasi-coherent (gs-modules with integrable connection relative to T. This connection is the Gauss-Martin connection on the El terms and on the abutment. In particular, the W filtration on the Rkf,(Q}/s(logD)) is horizontal for the Gauss-Manin connection. Proof. The product structure in the spectral sequence results from (4.0.1.1). The action of the Gauss-Manin connection on the spectral sequence may be seen directly as follows. The question being local on S, we may suppose S affine. Choose an affine open covering of X by suf- ficiently small coordinatized open sets V v as in (3.4.1). The filtration W of f2~/s(logD) induces a filtration W on the Cech bicomplex of quasi-coherent (gs-modules (4.0.2.0) W~ C~ t2}/s(logD)) dN,, C~ W~f21/s(logO) )...
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