# Generalisation de la theorie arithmetique des D-modules a la geometrie logarithmique

Abstract : This work aims at generalizing the arithmetic theory of D-modules to logarithmic geometry. We first define sheaves of differentials operators of level m. We describe those sheaves D(m) in local coordinates in the log-smooth case, just as Berthelot did in the non-logarithmic case. Then we study the action of the Frobenius homomorphism on D(m)-modules, and show that F* makes the level raise. Nevertheless, the descent theorem proved by Berthelot for usual schemes is not true in general for log-schemes. Thus we use the work by Lorenzon, which associates a canonical algebra A to a log-scheme, and we get an equivalence of categories between A x D(m) -modules and B x D(0) -modules (indexed). Finally we deduce from this equivalence that the cohomological dimension of D(m) is finite, when X is a smooth scheme over a field, and M is defined by a divisor with normal crossings.
Mots-clés :
Document type :
Theses
Domain :

https://tel.archives-ouvertes.fr/tel-00002545
Contributor : Claude Dumas-Montagnon <>
Submitted on : Wednesday, March 12, 2003 - 6:15:26 PM
Last modification on : Thursday, January 7, 2021 - 4:12:37 PM
Long-term archiving on: : Tuesday, September 7, 2010 - 4:08:12 PM

### Identifiers

• HAL Id : tel-00002545, version 1

### Citation

Claude Montagnon. Generalisation de la theorie arithmetique des D-modules a la geometrie logarithmique. Mathématiques [math]. Université Rennes 1, 2002. Français. ⟨tel-00002545⟩

Record views