Finiteness of de Rham cohomology in rigid analysis

For a large class of smooth dagger spaces-rigid spaces with overconvergent structure sheaf-we prove finite dimensionality of de Rham cohomology. This is enough to obtain finiteness of P Berthelot's rigid cohomology also in the nonsmooth case. We need a careful study of de Rham cohomology, in situations of semistable reduction.