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

Let k be a perfect field of characteristic p > 0, let Ad be the coordinate ring of the coordinate axes in affine d–space over k, and let Id be the ideal defining the origin. We evaluate the relative K–groups Kq(Ad,Id) in terms of p–typical Witt vectors of k. When d = 2 the result is due to Hesselholt, and for K2 it is due to Dennis and Krusemeyer. We also compute the groups Kq(Ad,Id) 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.

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algebraic $K$–theory, coordinate axes, topological Hochschild homology, topological cyclic homology, cyclotomic spectra
Mathematical Subject Classification 2010
Primary: 19D55
Secondary: 16E40
Received: 15 April 2019
Revised: 27 May 2020
Accepted: 15 June 2020
Published: 25 February 2021
Martin Speirs
Department of Mathematics
University of California, Berkeley
Berkeley, CA
United States