De Rham and infinitesimal cohomology in Kapranov's model for noncommutative algebraic geometry
arXiv:math/0102133
Abstract
The title refers to the nilcommutative or -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 -schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for - and -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 as a sum of copies of the usual (de Rham, infinitesimal) cohomology groups of the underlying commutative scheme (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.
arXiv comment38 pages, uses xy.tex for diagrams