#### Volume 22, issue 1 (2022)

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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 $R\left[{x}_{1},\dots ,{x}_{d}\right]∕{\left({x}_{1},\dots ,{x}_{d}\right)}^{2}$. 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 ${K}_{\ast }\left(R\left[{x}_{1},\dots ,{x}_{d}\right]∕{\left({x}_{1},\dots ,{x}_{d}\right)}^{2},\left({x}_{1},\dots ,{x}_{d}\right)\right)$ when $R$ is a perfect ${\mathbb{𝔽}}_{p}$–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