Infinitesimal K-theory

TL;DR

This paper investigates the fiber of the rational Jones-Goodwillie character map from K-theory to negative cyclic homology, describing it via sheaf cohomology and relating its homotopy groups to non-commutative infinitesimal hypercohomology.

Contribution

It provides a new description of the fiber of the rational Jones-Goodwillie character in terms of sheaf cohomology and establishes an isomorphism with non-commutative infinitesimal hypercohomology groups.

Findings

Fiber F described via sheaf cohomology.

Homotopy groups of F linked to non-commutative infinitesimal hypercohomology.

Isomorphism for n ≥ 1 between π_n(F) and H^{-n}_{inf}(A,K^ at).

Abstract

In this paper we study the fiber F of the rational Jones-Goodwillie character $$ F:=\hofiber(ch:K^\rat(A)@>>>HN^\rat(A)) $$ going from K-theory to negative cyclic homology of associative rings. We describe this fiber F in terms of sheaf cohomology. We prove that, for $n\ge 1$, there is an isomorphism: $$ \pi_n(F)\cong H^{-n}_{inf}(A,K^\rat) $$ between the homotopy of the fiber and the hypercohomology groups of $K^\rat$ on a non-commutative version of Grothendieck's infinitesimal site.