2022/09/16 by Stum, Bernard Le · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.07875
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.