Conjectures with wide scope on making this into formal statements were enunciated by Jean-Marc Fontaine, the resolution of which is called p-adic Hodge theory.
It is not possible in general to find similar cohomology groups with coefficients in Qp (or Zp, or Q, or Z) having reasonable properties.
The classic reason (due to Serre) is that if X is a supersingular elliptic curve, then its endomorphism ring is a maximal order in a quaternion algebra B over Q ramified at p and ∞.
If X had a cohomology group over Qp of the expected dimension 2, then (the opposite algebra of) B would act on this 2-dimensional space over Qp, which is impossible since B is ramified at p.[1] Grothendieck's crystalline cohomology theory gets around this obstruction because it produces modules over the ring of Witt vectors of the ground field.
The site Inf(X) is a category whose objects can be thought of as some sort of generalization of the conventional open sets of X.
The reason is roughly that in order to prove exactness of the de Rham complex, one needs some sort of Poincaré lemma, whose proof in turn uses integration, and integration requires various divided powers, which exist in characteristic 0 but not always in characteristic p. Grothendieck solved this problem by defining objects of the crystalline site of X to be roughly infinitesimal thickenings of Zariski open subsets of X, together with a divided power structure giving the needed divided powers.
We will work over the ring Wn = W/pnW of Witt vectors of length n over a perfect field k of characteristic p>0.
(More generally one can work over a base scheme S which has a fixed sheaf of ideals I with a divided power structure.)
If X is a scheme over k, then the crystalline site of X relative to Wn, denoted Cris(X/Wn), has as its objects pairs U→T consisting of a closed immersion of a Zariski open subset U of X into some Wn-scheme T defined by a sheaf of ideals J, together with a divided power structure on J compatible with the one on Wn.
The term crystal attached to the theory, explained in Grothendieck's letter to Tate (1966), was a metaphor inspired by certain properties of algebraic differential equations.