Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\mathscr{D}$-modules

Details Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\mathscr{D}$-modules Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\mathscr{D}$-modules

2011-05-29

arxiv.org/abs/1105.5796v1

Mathematics

Algebraic Geometry

Scientific paper

The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic $\ms{D}$-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincar\'{e} duality in the style of SGA4. As another application, we compare the push-forward as arithmetic $\ms{D}$-modules and the rigid cohomologies taking Frobenius into account. These theorems will lead us to an analog of "Weil II" and a product formula for $p$-adic epsilon factors.

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