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arXiv · 2205.06104

The localisation theorem for the $\mathrm{K}$-theory of stable $\infty$-categories

Fabian Hebestreit·Andrea Lachmann·Wolfgang Steimle
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Abstract

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\mathrm{K}$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\mathrm{K}$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and demonstrate that the cofinality theorem can be derived from the universal property alone.

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Fabian Hebestreit, Andrea Lachmann, Wolfgang Steimle. 2022-05-12. The localisation theorem for the $\mathrm{K}$-theory of stable $\infty$-categories. https://arxiv.org/abs/2205.06104

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