On the K–theory of coordinate axes in affine space

Speirs, Martin

doi:10.2140/agt.2021.21.137

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On the $K$–theory of coordinate axes in affine space

Martin Speirs

Algebraic & Geometric Topology 21 (2021) 137–171

DOI: 10.2140/agt.2021.21.137

Abstract

Let $k$ be a perfect field of characteristic $p > 0$ , let $A_{d}$ be the coordinate ring of the coordinate axes in affine $d$ –space over $k$ , and let $I_{d}$ be the ideal defining the origin. We evaluate the relative $K$ –groups $K_{q} (A_{d}, I_{d})$ in terms of $p$ –typical Witt vectors of $k$ . When $d = 2$ the result is due to Hesselholt, and for $K_{2}$ it is due to Dennis and Krusemeyer. We also compute the groups $K_{q} (A_{d}, I_{d})$ in the case where $k$ is an ind-smooth algebra over the rationals, the result being expressed in terms of algebraic de Rham forms. When $k$ is a field of characteristic zero this calculation is due to Geller, Reid and Weibel.

Keywords

algebraic $K$–theory, coordinate axes, topological Hochschild homology, topological cyclic homology, cyclotomic spectra

Mathematical Subject Classification 2010

Primary: 19D55

Secondary: 16E40

References

Publication

Received: 15 April 2019

Revised: 27 May 2020

Accepted: 15 June 2020

Published: 25 February 2021

Authors

Martin Speirs
Department of Mathematics University of California, Berkeley Berkeley, CA United States

Volume 21, issue 1 (2021)