Overcoherent arithmetic \(\mathcal D\)-modules. Application to \(L\)-functions.
type: Item
instance of: scholarly article
title: Overcoherent arithmetic \(\mathcal D\)-modules. Application to \(L\)-functions.
zbMATH Open document ID: 1129.14030
author: Daniel Caro
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116165
review text: This article is one piece of the author's program to construct a category of \(p\)-adic coefficients, attached to a scheme over a finite field \(k\), which is stable by the six Grothendieck operations: direct image (resp. exc. direct image), inverse image (resp. exc. inverse image), (external) tensor product, duality and local cohomological functors. In this review \(D^{\dagger}\) will denote the sheaf of arithmetic \(D\)-modules of Berthelot and \(V\) is a discrete valuation ring such that \(k\) is the residue field of \(V\). We will work with modules which are endowed with a Frobenius structure denoted by \(F\), that means \(D^{\dagger}\)-modules \(M\) endowed with a \(D^{\dagger}\)-linear isomorphism \(F^*M\simeq M\). Let us notice that some interesting results are valid in this article for arithmetic \(D\)-modules which are not endowed with such a structure. Conjecturally, the category of holonomic \(D^{\dagger}\)-modules of \textit{P. Berthelot} [``D-modules arithmétiques. II. Descente par Frobenius.'' Mém. Soc. Math. Fr., Nouv. Sér. 81 (2000; Zbl 0948.14017)] endowed with a Frobenius structure should be stable by the \(6\) operations but this result is not yet known. The approach here consists into defining a new subcategory of the derived category of arithmetic \(D^{\dagger}\)-modules with overconvergent singularities along a divisor, by forcing some property of stability by image inverse functors that holonomic \(F\)-\(D^{\dagger}\)-modules should conjecturally verify. This category is called the category of overcoherent \(F\)-\(D^{\dagger}\)-modules. In the case of a curve, it is known that an overcoherent \(D^{\dagger}\)-module is holonomic. In general, it is a consequence of Berthelot's conjectures that the category introduced here is the same as the category of holonomic \(F\)-\(D^{\dagger}\)-modules defined by ~Berthelot. As an example unit-root \(F\)-isocrystals are overcoherent. One application of this construction is that D.~Caro can define a categor…
reviewed by: Christine Noot-Huyghe
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 2162447
zbMATH Keywords: arithmetic \(\mathcal D\)-modules
zbMATH Keywords: overconvergent \(\mathcal D\)-modules
zbMATH Keywords: Grothendieck six operations
MaRDI profile type: Publication
OpenAlex ID: W2317054216
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DOI: 10.5802/AIF.2072
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2162447