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

TL;DR

This paper explores noncommutative and Poisson nilpotent thickenings of schemes, developing variants of de Rham and infinitesimal cohomology that reveal a Hodge decomposition linking noncommutative and classical cohomology theories.

Contribution

It introduces new variants of de Rham and infinitesimal cohomology for noncommutative and Poisson schemes, establishing a Hodge decomposition that connects these to classical cohomology.

Findings

Noncommutative variants admit Hodge decompositions.

Cohomology groups of noncommutative schemes relate to classical cohomology.

New proofs of classical results on de Rham and infinitesimal cohomology.

Abstract

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.