A generalization of Coleman's p-adic integration theory

We associate to a scheme X smooth over a p-adic ring a kind of cohomology group Hⁱ_fp(X, j). For proper X this cohomology has Poincaré duality hence Gysin maps and cycle class maps which are reasonably explicit. For zero-cycles we show that the cycle class map is given by Coleman integration. The cohomology theory H_fp is therefore interpreted as giving a generalization of Coleman's theory. We find an embedding H²ⁱ_syn(X, i) → H²ⁱ_fp(X, i) where H_syn is (rigid) syntomic cohomology. Our main result is an explicit description of the syntomic Abel-Jacobi map in terms of generalized Coleman integration.