Dr. Katz DEq Homework Solutions 72

Dr. Katz DEq Homework Solutions 72 - 72 N.M Katz F or each...

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72 N.M. Katz: For each Q-conjugacy class A of irreducible C-representations of G, the element ( P(A) der'" ~ P(Z) Z~A ofC [G] lies in fact in Q [G], and is an indecomposable central idempotent in Q [G]. Every indecomposable central idempotent in Q [G] is of this form. We have ( p(A)P(A,)=~P(A) if A=A' O if not. ( Let Hz be a local system of Z-modules of finite type on which G acts. We define the G-sub-local system ( P(A) H z de_~f'n the inverse image in H z of P(A)(Ho) ~ HQ. The canonical mapping of local systems ( (~ P(A) H z --~ H z A is a G-morphism and an isogeny, i.e., it becomes an isomorphism when tensored with Q. (4.4.1) If G acts on a family of mixed Hodge structures (H z, W,F) (meaning that its action on the local system Hz respects W and F), then each P(A) H z is a sub-family of mixed Hodge structures. If (H z, W, F) is a polarizable ( family of mixed Hodge structures, so is each sub-
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