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 nonconnective $K$-theory, topological Hochschild homology and topological cyclic homology as well as connective $K$-theory of stable $\infty$-categories, and generalizes several known formulas for algebraic $K$-theory of spaces or connective $K$-theory of ordinary rings, ring spectra, schemes and $\mathbb{𝕊}$-algebras.

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