Integral p-adic cohomology theories for open and singular varieties

For open and singular varieties in positive characteristic p we study the existence of an integral p-adic cohomology theory which is finitely generated, compatible with log crystalline cohomology and rationally compatible with rigid cohomology. We develop such a theory under certain assumptions of resolution of singularities in positive characteristic, by using cdp- and cdh-topologies. Without resolution of singularities in positive characteristic, we prove the existence of a good p-adic cohomology theory for open and singular varieties in cohomological degree 1, by using split proper generically \'etale hypercoverings. This is a slight generalisation of a result due to Andreatta--Barbieri-Viale. We also prove that this approach does not work for higher cohomological degrees.