In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic generic fiber.
The so-called rigid cohomology extends crystalline cohomology to the case of not necessarily smooth schemes. For separated schemes of finite type, rigid cohomology groups satisfy reasonable finiteness properties (in contrast with crystalline cohomology). So maybe it is not entirely unreasonable to ask: if we consider rigid cohomology, does p-adic Hodge theory make sense for families of schemes with singular special fiber?
