Trace methods for stable categories I: The linear approximation of algebraic K-theory

TL;DR

This paper explores algebraic K-theory and topological Hochschild homology within stable categories, establishing universal properties and a trace map that links K-theory to THH, advancing understanding of their relationship.

Contribution

It introduces the concept of laced categories and demonstrates universal properties of K-theory and THH, leading to a new trace map and a generalization of Dundas-McCarthy's result.

Findings

Constructed a trace map from laced K-theory to THH.

Showed THH as the first Goodwillie derivative of laced K-theory.

Generalized Dundas-McCarthy's identification of stable K-theory.

Abstract

We study algebraic K-theory and topological Hochschild homology in the setting of bimodules over a stable category, a datum we refer to as a laced category. We show that in this setting both K-theory and THH carry universal properties, the former defined in terms of additivity and the latter via trace properties. We then use these universal properties in order to construct a trace map from laced K-theory to THH, and show that it exhibits THH as the first Goodwillie derivative of laced K-theory in the bimodule direction, generalizing the celebrated identification of stable K-theory by Dundas-McCarthy, a result which is the entryway to trace methods.