Trace methods for stable categories I: The linear approximation of algebraic K-theory
Algebraic Topology
2026-03-03 v2 K-Theory and Homology
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.
Cite
@article{arxiv.2411.04743,
title = {Trace methods for stable categories I: The linear approximation of algebraic K-theory},
author = {Yonatan Harpaz and Thomas Nikolaus and Victor Saunier},
journal= {arXiv preprint arXiv:2411.04743},
year = {2026}
}