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The fundamental theorem of localizing invariants

Victor Saunier

Vol. 8 (2023), No. 4, 609–643
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 -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 nonconnective K-theory, topological Hochschild homology and topological cyclic homology as well as connective K-theory of stable -categories, and generalizes several known formulas for algebraic K-theory of spaces or connective K-theory of ordinary rings, ring spectra, schemes and 𝕊-algebras.

Keywords
$K$-theory, Bass–Heller–Swan, localizing invariant, THH, stable infinity-category
Mathematical Subject Classification
Primary: 19D35
Secondary: 18F25, 19D10
Milestones
Received: 18 April 2023
Revised: 5 October 2023
Accepted: 1 November 2023
Published: 6 December 2023
Authors
Victor Saunier
Laboratoire Analyse, Géométrie et Applications
CNRS - Université Sorbonne Paris Nord
99 avenue Jean Baptiste Clément
93430 Villetaneuse
France