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Dr. Katz DEq Homework Solutions 58

Dr. Katz DEq Homework Solutions 58 - 58 N.M Katz w hose...

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58 N.M. Katz: whose assosciated graded is given by gr, w C~ V~},t2~/s(log D)) ( = ~ C~ .... Di,/S)" il<'"<in The spectral sequence ( is the spectral sequence of the filtered complex of quasi-coherent (~s-modules obtained from the filtered bicomplex ( by totalization. The (not necessarily integrable) T-connection V on the Cech bicomplex C ~ ({ Vv}, I2~/s (log D)) constructed in (3.4.1) preserves the filtration W. This provides a T-connection on the spectral sequence (, which is that of Gauss-Manin on the abutment. To see that it is that of Gauss-Manin on the E1 term, we simply note that on the assosciated graded for W, ( . C ~({ V~ n Dil n... n Di.}, ~-2~)il ..... Di,/S) it induces the same T-connection which the method of (3.4.1) would construct, viewing the affine open sets Vv n Dil n... n Din as a coordinatized open covering of the smooth S-scheme Di~n...nDi, (the coordinates being those on V~ whose restriction to Di~ n.-.n Di, do not vanish). Of course, the truth of (4.0.2) could also be perceived by "pure
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