  Given an associative unital algebra $A$ over a perfect field $k$ of odd
positive characteristic, we construct a non-commutative generalization of the
Cartier isomorphism for $A$. The role of differential forms is played by
Hochschild homology classes, and de Rham diferential is replaced with the
Connes-Tsygan differential.

Algebraic Geometry;K-Theory and Homology