On the algebraic K–theory of double points

Riggenbach, Noah

doi:10.2140/agt.2022.22.373

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On the algebraic $K$–theory of double points

Noah Riggenbach

Algebraic & Geometric Topology 22 (2022) 373–403

DOI: 10.2140/agt.2022.22.373

Abstract

We use trace methods to study the algebraic $K$ –theory of rings of the form $R [x_{1}, \dots, x_{d}] ∕ {(x_{1}, \dots, x_{d})}^{2}$ . 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 $K_{*} (R [x_{1}, \dots, x_{d}] ∕ {(x_{1}, \dots, x_{d})}^{2}, (x_{1}, \dots, x_{d}))$ when $R$ is a perfect $𝔽_{p}$ –algebra. We conclude with some other notable computations, including some rings which are not quite of the above form.

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

Volume 22, issue 1 (2022)