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70
N.M. Katz:
By (4.2.2.3), (4.3.1), and "Riemann's existence theorem" (cf. [36] and
the appendix) we have (4.3.3.1) ~ (4.3.3.2).
To see that (4.3.3.2) ~ (4.3.3.3), 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 (4.3.3.3)~ (4.3.3.0), we first remark that it
suffices to prove that (4.3.3.0) holds after any base change ~o: S'* S by a
finite 6tale morphism, because (4.3.3.0) is of a differential nature, hence
local on S for the 6tale topology. The hypothesis (4.3.3.3) 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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This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.
 Fall '11
 NormanKatz

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