Over convergent \(F\)-isocrystals and differential overcoherence. (\(F\)-isocristaux surconvergents et surcohérence différentielle.) (English) Zbl 1203.14025

This article is part of author’s successful attempt to build a category of arithmetic coefficients for varieties in car \(p>0\), stable by Grothendieck’s six operations (among which inverse images, direct images, tensor product, duality). The categories contructed by Caro are subcategories of arithmetic \(D\)-modules in the sense of P. Berthelot [\(p\)-adic cohomology and arithmetic applications (II). Paris: Société Mathématique de France. Astérisque 279, 1–80 (2002; Zbl 1098.14010)].
To achieve this goal, D. Caro first contructed a category of overcoherent arithmetic \(D\)-modules with Frobenius structure [J. Math. Sci., Tokyo 16, No. 1, 1–21 (2009; Zbl 1213.14041)], a priori stable by direct, inverse images and local cohomological functors, and then a subcategory of the previous one: the category of overholonomic \(D\)-modules. An essential step in his constructions is to prove that overconvergent \(F\)-isocrystals, which can be seen as elementary pieces in this theory, are overcoherent. This fact is the aim of this article, and is established for a variety \(Y\) smooth over a perfect field \(k\) of car \(p>0\). Caro proves moreover that complexes of arithmetic \(F\)-\(D\)-modules with bounded and overcoherent cohomology can be unscrewed into overconvergent \(F\)-isocrystals.


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