Dr. Katz DEq Homework Solutions 110

Dr. Katz DEq Homework Solutions 110 - 110 N.M. Katz:...

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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 T-connection 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 k-valued 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 k-valued 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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