#### Volume 21, issue 1 (2021)

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

### Martin Speirs

Algebraic & Geometric Topology 21 (2021) 137–171
##### 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 ${K}_{q}\left({A}_{d},{I}_{d}\right)$ 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 ${K}_{q}\left({A}_{d},{I}_{d}\right)$ 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
Primary: 19D55
Secondary: 16E40
##### 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