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110
N.M. Katz:
(7.4.0.6)
the necessary and sufficient condition for the invertible sheaf
~)s(,F ni
[P/])
to admit an integrable Tconnection is that ,F nl
=0
in A.
(7.4.1.0)
Suppose further that A is a principal ideal domain, with fraction
field k, and that S admits a
section
over A. Because
S/A
is projective, and
A is principal, any kvalued point of
Sk
extends to a unique section of S
over A, so it is the same to assume that
S k
has a rational point.
(7.4.1.1)
Then the Picard scheme
Pies/A
exists, is an extension of ZA
by the abelian subscheme PiC~/A, and its formation commutes with
arbitrary change of base A* A'. Because PiCs~ is
projective,
and A is
principal, any kvalued point of Pic~k/k = the Jacobian of
Sk
extends to
a unique section of Pic~
over A.
(7.4.1.2)
Because A is principal and
S/A
has a section, we have
(7.4.1.3)
HI(S, C*)
~ ~ Pics/A(A ).
We define Hi(S, t2~/r) 0 to be the inverse image of
Pic~
under
the canonical mapping
(7.4.1.4)
H I
(S, f2~/r) * H 1 (S, ~)~) ~ Pies~ A (A).
The long cohomology sequence (7.4.0.1) gives a short exact sequence
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 Fall '11
 NormanKatz

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