Non-commutative crystalline cohomology

Let $X$ be a scheme over $F_p$. The algebraic de Rham cohomology has the following striking property: for any two liftings $X_1$, $X_2$ of $X$ over $Z/p^nZ$ their de Rham cohomology $H_dR(X_1/Z/p^n)$ and $H_dR(X_2/Z/p^n)$ are canonically isomorphic. This leads to the definition of crystalline cohomology – a cohomology theory which assigns to any scheme $X$ a $Z/p^n$-module which is canonically isomorphic to the de Rham cohomology of any lifting(if it exists) and gives a completely new object if there are no lifting.
I will discuss a non-commutative analog of this construction. Namely, considering periodic cyclic homology of a $DG$ algebra as a non-commutative analog of the de Rham cohomology, non-commutative crystalline cohomology will be a functor which assigns to a $DG$ algebra $A$ over $F_p a Z/p^n$ module which is canonically isomorphic to the periodic cyclic homology of any lifting of $A$ over $Z/p^n$. This is a joint work with Vadim Vologodsky.