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Zbl 1165.14305
Caro, Daniel
Comparing the sheaves of overconvergent isocrystals. (Comparaison des facteurs duaux des isocristaux surconvergents.) (English)
[J] Rend. Semin. Mat. Univ. Padova 114, 131-211 (2005). ISSN 0041-8994

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 $\cal D$-modules [cf. for example {\it 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 $\cal D$-modules (the holonomic or overholonomic ones) are stable under five of Grothendieck's six operations. The article under review is part of that series.\par Let $\germ{X}$ be a smooth formal scheme, $\text{sp}$ the specialization map, $Z$ a divisor on the special fiber $X$ of $\germ{X}$, $E$ an isocrystal on $X\setminus Z$ overconvergent along $Z$, $E^\vee$ its dual and $\Bbb{D}_Z^\dagger$ the $\cal D$-module dual. The main result is that there is an isomorphism compatible with the Frobenius map: $$\Bbb{D}_Z^\dagger(\cal O_{\germ{X}} ({}^\dagger Z)_{\Bbb{Q}}) \otimes_{\cal O_{\germ{X}} ({}^\dagger Z)_{\Bbb{Q}}} \text{sp}_*(E^\vee) \simeq\Bbb{D}_Z^\dagger(\text{sp}_*(E)).$$ This result, which gives a $\cal{D}$-module theoretical interpretation of the dual of an overconvergent isocrystal, is a $p$-adic analogue of a characteristic zero result of Berthelot.
[Laurent Berger (Lyon) (MR2207865)]

MSC 2000:: *14F10 Special sheaves
14F30 p-adic cohomology
32C38 Sheaves of differential operators (analytic spaces)

Citations: Zbl 1098.14010

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