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Dr. Katz DEq Homework Solutions 70

Dr. Katz DEq Homework Solutions 70 - 70 N.M Katz B...

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70 N.M. Katz: By (, (4.3.1), and "Riemann's existence theorem" (cf. [36] and the appendix) we have ( ~ ( To see that ( ~ (, notice that (4.3.3,2) implies in particular that the local system (~oan)*(R"n."(C)) is constant on (S') an, i.e., iso- morphic to (Cb-)(s,),n. Thus, both q~*(R"f.(O~/s(logD)),V) and (((gs,)b",d) are coherent (gs,-modules with integrable connections having regular singular points (cf. 4.1) and they both give rise via (4.1.0) to isomorphic local systems on (S') ~". Hence they are isomorphic. In order to prove that ( (, we first remark that it suffices to prove that ( holds after any base change ~o: S'-* S by a finite 6tale morphism, because ( is of a differential nature, hence local on S for the 6tale topology. The hypothesis ( implies that the local system R" n, n (C) becomes constant on a finite 6tale covering, hence also R" n.n(Q) and R" n.n(z)/torsion become constant. As the torsion in R"n,'(Z) is a local system of finite groups, it follows that R'n.'(Z) becomes constant on a finite 6tale covering ~0': S"~ S, After making
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