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arXiv · 2209.05818

The Fundamental Theorem of Localizing Invariants

Victor Saunier
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Abstract

We prove a generalization of the fundamental theorem of algebraic K-theory for Verdier-localizing functors by extending the proof for algebraic K-theory of spaces to the realm of stable $\infty$-categories. The formula behaves much better for Karoubi-localizing functors, the Verdier-localizing invariants which are additionally invariant under idempotent completion. This general fundamental theorem specializes to new formulas in the context of non-connective K-theory, topological Hochschild homology and topological cyclic homology as well as connective K-theory of arbitrary ring spectra, and generalizes several known formulas for algebraic K-theory of spaces or connective K-theory of ordinary rings, schemes and $\mathbb{S}$-algebras.

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Victor Saunier. 2022-09-13. The Fundamental Theorem of Localizing Invariants. https://doi.org/10.2140/akt.2023.8.609

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