UC Berkeley

We extend le Stum's construction of the overconvergent site to algebraic stacks. We prove that etale morphisms are morphisms of cohomological descent for finitely presnted crystals on the overconvergent site. Finally, using the notion of an open subtopos of SGA4, we define a notion of overconvergent cohomology supported in a closed substack and show that it agrees with the classical notion of rigid cohomology supported in a closed subscheme.