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Comparing the sheaves of overconvergent isocrystals. (Comparaison des facteurs duaux des isocristaux surconvergents.) (English) Zbl 1165.14305

Summary: This article fits within the general program of defining a good category of \(p\)-adic coefficients, a program started by Berthelot, who introduced the notion of arithmetic \(\mathcal D\)-modules [cf. for example P. Berthelot, Astérisque No. 279, 1–80 (2002; Zbl 1098.14010)]. The author aims to prove in a series of papers that certain categories of \(\mathcal D\)-modules (the holonomic or overholonomic ones) are stable under five of Grothendieck’s six operations. The article under review is part of that series.
Let \(\mathfrak{X}\) be a smooth formal scheme, \(\text{sp}\) the specialization map, \(Z\) a divisor on the special fiber \(X\) of \(\mathfrak{X}\), \(E\) an isocrystal on \(X\setminus Z\) overconvergent along \(Z\), \(E^\vee\) its dual and \(\mathbb{D}_Z^\dagger\) the \(\mathcal D\)-module dual. The main result is that there is an isomorphism compatible with the Frobenius map: \[ \mathbb{D}_Z^\dagger(\mathcal O_{\mathfrak{X}} ({}^\dagger Z)_{\mathbb{Q}}) \otimes_{\mathcal O_{\mathfrak{X}} ({}^\dagger Z)_{\mathbb{Q}}} \text{sp}_*(E^\vee) \simeq\mathbb{D}_Z^\dagger(\text{sp}_*(E)). \] This result, which gives a \(\mathcal{D}\)-module theoretical interpretation of the dual of an overconvergent isocrystal, is a \(p\)-adic analogue of a characteristic zero result of Berthelot.

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F30 \(p\)-adic cohomology, crystalline cohomology
32C38 Sheaves of differential operators and their modules, \(D\)-modules

Citations:

Zbl 1098.14010
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References:

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