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.

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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

Volume 22, issue 1 (2022)