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

Yonatan Harpaz; Thomas Nikolaus; Victor Saunier

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arXiv · 2411.04743

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

Yonatan Harpaz·Thomas Nikolaus·Victor Saunier

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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.

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Yonatan Harpaz, Thomas Nikolaus, Victor Saunier. 2024-11-07. Trace methods for stable categories I: The linear approximation of algebraic K-theory. https://arxiv.org/abs/2411.04743

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