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Dr. Katz DEq Homework Solutions 40

Dr. Katz DEq Homework Solutions 40 - a b F~b s(R f(Qx/s(log...

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40 N.M. Katz: is endowed with the action of the Gauss-Manin connection, and that on the E2 terms, the p-curvature is zero. This implies that the p-curvature also vanishes on the Eo~ terms, and hence that (3.1.2) ~(F~o,R"f,(gJ~,/s(logD)))cF*~(fJ~/r)| +1 R"f, (gJ~r (log D)). Passing to the associated graded, there is an induced mapping, again denoted r (3.1.3) ~J: gr~oo, (R"f, (O~/s(log O))) --* F~* s (~2s~/7-) | gr~ooo (R"f, (~J~?s(log D))). Our main technical result 3.2 identifies this mapping, under suitable hypotheses with a twisted form of the Kodaira-Spencer mapping ( 3,2. Theorem. Under the hypotheses of (2.3.2), the diagram below is commutative g~oo,(Ra+Of,(CJ~./s(logD))) ~ , F*~(fJ~/r)| ) conE~b 17. [O 1 ]t'~ g'a+l,b-1 a abs ~a'~S/T] ~ con~ eong~,b b'* ~O1 "~t~ tT'a+l,b-1 .t abs I,aaS/T] ~ eon~L' 2 ~k
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Unformatted text preview: a b -- F~b s (R f, (Qx/s (log D))) ( 1)b+'F~*b~(p~ Fa~ (~/T) | F*~ (R ~ + ~f, (~b/j (log D))) bs s ~ Fa*bs Rof, D)) (- '' . ... | g o +'/, (log D)) in which p is the cup-product with the Kodaira-Spencer class (1.1.3), viewed as a global section over S of 1 1 Os/r| R f, (Dero(X/S)). Before proceeding to the proof, we will recall some basic facts about the modular representation theory of finite groups of order prime to p, and then restate 3.2 "with a group of operators ". (3.2.1) Let G be a finite group of order prime to p, and k a field of characteristic p. Let V be a finite-dimensional k-space on which G acts as a group of k-automorphisms, through a homomorphism Z: G ~ GL(V). Denoting by F, bs: k~ k the absolute Frobenius endomorphism of k, the representation Z (v) of G on F.'~,s(V)=V| (where k is a module over k...
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  • Fall '11
  • NormanKatz
  • finite group, Representation theory of finite groups, Modular representation theory, absolute Frobenius endomorphism, main technical result

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