D-modules arithmétiques II

Résumé du livre

In algebraic geometry, regardless of the characteristic, the theory of modules over suitable rings of differential operators, generically called $\mathcal D$-modules, is an essential tool in the study of de Rham cohomology and other theories derived from it (crystalline and rigid cohomologies). In this memoir, the author studies the particular properties of the action of a lifting of the Frobenius morphism on the category of $\mathcal D$-modules when the base is a scheme annihilated by a power of a fixed prime $p$ or a $p$-adic formal scheme. The main result is a descent theorem for the Frobenius morphism, allowing to reduce the study of modules endowed with an action of usual differential operators of order $\leq p^{m}$ to that of modules endowed with an action of derivations. The author proves the compatibility of this descent with all cohomological operations from $\mathcal D$-module theory, which shows that any $\mathcal D$-module of geometric origin can be endowed, in a suitable sense, with a natural Frobenius action. Some applications are included. Text is in French.