Periodic cyclic homology as sheaf cohomology

TL;DR

This paper establishes a connection between noncommutative infinitesimal cohomology and periodic cyclic cohomology, extending classical results relating sheaf cohomology and de Rham cohomology to a noncommutative setting.

Contribution

It introduces a noncommutative analogue of the infinitesimal site and proves that its cohomology corresponds to periodic cyclic cohomology, generalizing Grothendieck's classical theorem.

Findings

Cohomology of the structure sheaf modulo commutators equals periodic cyclic cohomology.

Computed noncommutative infinitesimal cohomology for various sheaves.

Showed hypercohomology with K-theory coefficients relates to the Jones-Goodwillie character.

Abstract

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example we show that hypercohomology with coefficients in $K$-theory gives the fiber of the Jones-Goodwillie character which goes from $K$-theory to negative cyclic homology.