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Dr. Katz DEq Homework Solutions 109

Dr. Katz DEq Homework Solutions 109 - ~ I'P(S(9 dlog Hi(S...

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Algebraic Solutions of Differential Equations 109 the zero cocycle (0, 1). If a d.t.k, to dies in H1(S, f2~/k), there exists an open covering q/i of S, and on each a# i there exists a closed coi~F(~[i, ~Qls/T) ( a meromorphic function gie K* an invertible function hieF(qli, (9*) such that ( co=coi+dgi/g i on q/i coi=dhl/hi on q/i gj/gi = hi/h2 on 0///n q/j. Then co = d(g i hi)/gi hi, and gi hi = gj hj is a global meromorphic function, so co is logarithmic. Q.E.D. 7.4. The Case of Curves (7.4.0) Let T be affine, say T= Spec(A), and S a projective and smooth curve over T with geometrically connected fibres. Then f2*/r is just the two-term complex ( (9~ dl~ Passing to hypercohomology, we have a long exact sequence 0 -~ F(S, f2~/r)-~ Ha(S, f2*/r) (
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Unformatted text preview: ~ I'P(S, (9*) dlog , Hi(S, f~s/r) (the left hand zero because F(S, (9*)=F(T, (9*) is annihilated by d log). The trace morphism (of. [15, 42]) defines a functorial isomorphism ( H 1(S, Qs~/r) ~, a = F(T, (9r)- ( If P is a section of S/T, its image IPI is a divisor in S which is smooth over T. The inverse of its sheaf of ideals is an invertible sheaf on S, noted classically (gs([P]). If P1, .-., Pr are sections, and n 1, . .., nr integers, we define the invertible sheaf ( (9 s (Z n i [P~]) = (gs ([P1])|174 ... | (9s ([P~])| The composite mapping trace ( HI(S, (9,) dlog ) HI(S, ~r-~IS/T) ~~,A allows us to attach to the class in Hi(S, (9*) of Os(Z ni [Pi]) an element of A, which is none other than the image in A of the integer 2; ni. Thus...
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