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Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic \(\mathcal D\)-modules. (English) Zbl 1327.14101

Summary: The aim of this paper is to compute the Frobenius structures of some cohomological operators of arithmetic \({\mathcal D}\)-modules. To do this, we calculate explicitly an isomorphism between canonical sheaves defined abstractly. Using this calculation, we establish the relative Poincaré duality in the style of [Théorie des topos et cohomologie étale des schémas (SGA 4). Un séminaire dirigé par M. Artin, A. Grothendieck, J. L. Verdier. Avec la collaboration de P. Deligne, B. Saint-Donat. Tome 3. Exposés IX à XIX. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0245.00002)]. As another application, we compare the push-forward as arithmetic \({\mathcal D}\)-modules and the rigid cohomologies taking Frobenius into account. These theorems will be used to prove “\(p\)-adic Weil II” and a product formula for \(p\)-adic epsilon factors.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
12H25 \(p\)-adic differential equations

Citations:

Zbl 0245.00002
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References:

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