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Devissage of the \(F\)-complexes of arithmetic \(\mathcal D\)-modules in overconvergent \(F\)-isocrystals. (Dévissages des \(F\)-complexes de \(\mathcal D\)-modules arithmétiques en \(F\)-isocristaux surconvergents.) (French) Zbl 1114.14011

Let \(V\) be a discrete valuation ring of characteristics \((0,p)\), \(K\) its fraction field, \(k\) the residue field, \(\mathcal{P}\) a smooth proper formal scheme over \(\text{Spf}(V)\), \(P \) the special fiber of \(\mathcal{P}\), \(U\) the complementary of a divisor, and \(Y\) a smooth closed scheme of \(U\). In the \(\ell\)-adic context, it is known for a long time that constructible \(\mathbb{Q}_\ell\)-adic sheaves over \(Y\) (\(\ell\neq p\)) can be decomposed “by devissage” as direct images along strata of smooth \(\ell\)-adic sheaves.
The aim of this paper is to show that overholonomic arithmetic \(\mathcal{D}\)-modules over \(\mathcal{P}\) with a Frobenius admit a decomposition along strata of \(Y\), as direct images of overconvergent \(F\)-isocrystals. This result is important and has several applications.
One application is to prove that the category of overholonomic \(\mathcal{D}\)-modules, endowed with a Frobenius, is stable by tensor products. This category introduced by the author is stable by other cohomological operations, as proved in [D. Caro, \(\mathcal{D}\)-modules arithmetiques surholonomes, preprint] and unit-root \(F\)-isocrystals are overholonomic \(\mathcal{D}\)-modules.
A second important application of this theorem consists to give a \(p\)-adic Weil II statement based on the analogous statement of Kedlaya for overconvergent \(F\)-isocrystals [K. S. Kedlaya, Compos. Math. 142, No. 6, 1426–1450 (2006; Zbl 1119.14014)].
A third important application is that if \(\mathcal{U}\) is a smooth lifting of \(U\), then the restriction to \(\mathcal{U}\) is fully faithfull for “devissable” \(F\)-complexes of arithmetic \(\mathcal{D}\)-modules. An important part of the paper is devoted to the definition of the direct image over \(\mathcal{P}\) of an overconvergent \(F\)-isocrystal, as an arithmetic \(\mathcal{D}\)-modules, by a functor \(\text{sp}_+\). This was done in the simple case where \(Y\) is the special fiber of the complementary of a divisor in \(\mathcal{P}\) by P. Berthelot [in: \(p\)-adic analysis, Proc. Int. Conf., Trento/Italy 1989, Lect. Notes Math. 1454, 80–124 (1990; Zbl 0722.14008)] and by the author [Bull. Soc. Math. Fr. 137, No. 4, 453–543 (2009; Zbl 1300.14021), preprint arXiv:math/0500422]. The construction here is more general and works in the case of a smooth compactification. It gives the previous construction of the author over an open dense subset of \(Y\) thanks to de Jong’s desingularisation theorem and as exspected, the de Rham cohomology of the direct image by sp\(_+\) of an overconvergent \(F\)-isocrystal is equal to the rigid cohomology of the \(F\)-isocrystal. This construction is an important tool to define “devissable” \(F\)-complexes of arithmetic \(\mathcal{D}\)-modules. This is a conjecture that such complexes are in fact overholonomic, which is true if standard conjectures of Berthelot over holonomy of arithmetic \(\mathcal{D}\)-modules with Frobenius structure, are true.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F40 de Rham cohomology and algebraic geometry
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