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Algebraic Solutions of Differential Equations
77
Then after the change of base
Spec(C)~
T (cf. the diagram
(5.1.1.0))
each of the coherent sheaves with integrable connection on Sc
(P(x) R"fc.(t2~Cc/sc(log
Oc)) V),
x~A
becomes trivial on a finite &ale covering q9: S' ~
Sc, i, e.,
becomes isomorphic
to
(((gs,) b"(z),
d), where b,(z) is the rank of P(X) R'f,(f~/s(log D)) over the
generic point of S.
5.4. The Birational Point of View
(5.4.0)
Let Sc be a connected smooth Cscheme, and (Me, 17c) a locally
free sheaf of finite rank with integrable connection on Sc.
(5.4.1)
Remark.
In order that there exist a finite 6tale covering of Sc on
which (Me, Vc) becomes trivial (as locally free sheaf with integrable
connection), it is necessary and sufficient that there exist a finite exten
sion K of the function field L of Sc such that (Me, Vc) K is trivial (i. e.,
so that
Mc
is spanned over K by horizontal sections).
(5.4.2)
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 Fall '11
 NormanKatz

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