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Convolution and Equidistribution by Nicholas M. Katz

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CHAPTER 3

Fibre Functors

At this point, we introduce the fibre functor suggested by Deligne. The proof that it is in fact a fibre functor is given in the Appendix.

Theorem 3.1. (Deligne) Denoting by

j0 : Gm/kA1/k

the inclusion, the construction

M image H0(A1/k, j0!M)

is a fibre functor on the Tannakian category P.

For any Kummer sheaf Lχ on Gm/k, the operation M image MLχ is an autoequivalence of P with itself as Tannakian category. So we get the following corollary.

Corollary 3.2. For any Kummer sheaf Lχ on Gm/k, the construction

M H0(A1/k, j0!(

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