De Rham and infinitesimal cohomology in Kapranov's model for noncommutative algebraic geometry

The title refers to the nilcommutative or $NC$-schemes introduced by M. Kapranov in math.AG/9802041. The latter are noncommutative nilpotent thickenings of commutative schemes. We consider also the parallel theory of nil-Poisson or $NP$-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for $NC$- and $NP$-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck. It turns out that each of these noncommutative variants admits a kind of Hodge decomposition which allows one to express the cohomology groups of a noncommutative scheme $Y$ as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme $X$ (Theorems 6.2, 6.5, 6.8). As a byproduct we obtain new proofs for classical results of Grothendieck (Corollary 6.3) and of Feigin-Tsygan (Corollary 6.9) on the relation between de Rham and infinitesimal cohomology and between the latter and periodic cyclic homology.