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On the algebraic $K$–theory of double points

Noah Riggenbach

Algebraic & Geometric Topology 22 (2022) 373–403
Abstract

We use trace methods to study the algebraic K–theory of rings of the form R[x1,,xd](x1,,xd)2. We compute the relative p–adic K groups for R a perfectoid ring. In particular, we get the integral K groups when R is a finite field, and the integral relative K groups K(R[x1,,xd](x1,,xd)2,(x1,,xd)) when R is a perfect 𝔽p–algebra. We conclude with some other notable computations, including some rings which are not quite of the above form.

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Keywords
K–theory, topological cyclic homology, perfectoid ring, K–theory of singular schemes
Mathematical Subject Classification
Primary: 19D50
Secondary: 14G45, 19D55, 55P42, 55P91
References
Publication
Received: 7 August 2020
Revised: 11 January 2021
Accepted: 12 February 2021
Published: 26 April 2022
Authors
Noah Riggenbach
Department of Mathematics
Indiana University
Bloomington, IN
United States
Department of Mathematics
Northwestern University
Evanston, IL
United States