In the article in which Grothendieck first proposed $p$-adic cohomology theories, he gave as motivation for such theories the study of torsion phenomenon. Crystalline cohomology gives a theory with all the desirable formal properties (including finiteness) for proper smooth schemes over a field of positive characteristic $p>0$, but the only theories valid for more general schemes have coefficients in a field of characteristic $0$, so that torsion is invisible. In fact a “integral” theory with valid for arbitrary separated schemes of finite type over a field has yet to be constructed. I will first review such partial results as are available today, and then show that no integral $p$-adic cohomology theory with “reasonable” finiteness properties can be compatible with finite etale descent. This is joint work with Tomoyuki Abe.
