The Fundamental Theorem of Localizing Invariants

TL;DR

This paper generalizes the fundamental theorem of algebraic K-theory to Verdier-localizing invariants within stable ∞-categories, providing new formulas for various homology theories and K-theory of ring spectra.

Contribution

It extends the fundamental theorem to Verdier-localizing functors and introduces formulas applicable to non-connective K-theory, THH, and TC, broadening the scope of algebraic K-theory.

Findings

Generalized fundamental theorem for Verdier-localizing invariants

Derived new formulas for non-connective K-theory, THH, and TC

Unified several known formulas in algebraic K-theory context

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.