Rigid cohomology of locally noetherian schemes Part 2 : Crystals

Abstract: We introduce the general notions of an overconvergent site and a constructible crystal on an overconvergent site. We show that if $V$ is a geometric materialization of a locally noetherian formal scheme $X$ over an analytic space $O$ defined over $\mathbb Q$, then the category of constructible crystals on $X/O$ is equivalent to the category of constructible modules endowed with an overconvergent connection on the tube $\,]X[_V$ of $X$ in $V$. We also show that the cohomology of a constructible crystal is then isomorphic to the de Rham cohomology of its realization on the tube $\,]X[_V$. This is a generalization of rigid cohomology. Finally, we prove universal cohomological descent and universal effective descent with respect to constructible crystals with respect to the $h$-topology. This encompass flat and proper descent and generalizes all previous descent results in rigid cohomology.