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Dr. Katz DEq Homework Solutions 108

Dr. Katz DEq Homework Solutions 108 - 108 N.M Katz(7.2.3 R...

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108 N.M. Katz: (7.2.3) Remark. ~, depends additively upon the invertible sheaf with integrable connection (~, V). 7.3. When T is (the spectrum of) a field k of characteristic zero, the group Hl(S, t2~/r) has a classical interpretation. Recall that a mero- morphic closed one-form co on S is said to be a differential of the third kind if there exists an open covering {q/i} of S, and on each q/i a closed holomorphic one form (Diff_F(d~i,~l/T) and a non-zero meromorphic function gi on q/i, such that ( co=coiq-dgi/gi onq/i, dcoi=O. Now define f~j = gj/gi, a meromorphic function on q/i n q/j. In fact, fiijEz F(O~Zi ("3 q/j, (Q~ ), because on q/i n q/j ( dfii/fij = dgj/gj- d gi/gi = coi - coj so that dfij/fij is holomorphic on q/i c3 q/j. Because we are in characteristic zero, this implies that fij and 1/f~j are holomorphic on q/i ~ q/j. Thus we have assosciated to a differential of the third kind co on S a Cech 1-cocycle (coi,flj) for the complex I2~/T. If, on the same covering q/i,
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