The localisation theorem for the K -theory of stable ∞-categories

Fabian Hebestreit; Andrea Lachmann; Wolfgang Steimle

doi:10.1017/prm.2023.35

The localisation theorem for the K -theory of stable ∞-categories

Fabian Hebestreit, Andrea Lachmann, Wolfgang Steimle

Mathematical Science

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Abstract

We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic -theory of stable -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 -theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of -theory.

Original language	English
Pages (from-to)	1-37
Number of pages	37
Journal	Proceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume	153
Issue number	4
Early online date	20 Jul 2023
DOIs	https://doi.org/10.1017/prm.2023.35
Publication status	E-pub ahead of print - 20 Jul 2023

Bibliographical note

Funding Information:
We heartily thank Dustin Clausen, from whom we first learned a clean proof of the localisation theorem, for several very helpful discussions. We further thank Ferdinand Wagner for the permission to use some of his pretty TikZ diagrams in this note. During the preparation of this manuscript, F. H. was a member of the Hausdorff Center for Mathematics at the University of Bonn funded by the German Research Foundation (DFG) and furthermore a member of the cluster ‘Mathematics Münster: Dynamics-Geometry-Structure’ at the University of Münster (DFG grant nos. EXC 2047 390685813 and EXC 2044 390685587, respectively). F. H. acknowledgesthe Mittag-Leffler Institute for its hospitality during the research programme ‘Higher algebraic structures in algebra, topology and geometry’, supported by the Swedish Research Council under grant no. 2016-06596. A. L. was supported by the Research Training Group ‘Algebro-Geometric Methods in Algebra, Arithmetic and Topology’ at the University of Wuppertal (DFG grant no. GRK 2240) and W. S. by the priority programme ‘Geometry at Infinity’ (DFG grant no. SPP 2026) at the University of Augsburg.

Publisher Copyright:
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.

Keywords

algebraic K-theory

Access to Document

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Cite this

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abstract = "We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic -theory of stable -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 -theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of -theory.",

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The localisation theorem for the K -theory of stable ∞-categories

Abstract

Bibliographical note

Keywords

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