On The Algebraic $K$-Theory of Double Points

TL;DR

This paper employs trace methods to compute algebraic K-theory groups of specific rings involving double points, providing explicit calculations for perfectoid rings, finite fields, and perfect $ ext{F}_p$-algebras.

Contribution

It introduces new computations of algebraic K-theory for rings with nilpotent elements, extending known results to perfectoid and perfect $ ext{F}_p$-algebras.

Findings

Computed relative p-adic K-groups for perfectoid rings.

Derived integral K-groups for finite fields.

Calculated relative K-groups for perfect $ ext{F}_p$-algebras.

Abstract

In this paper, we use trace methods to study the algebraic $K$-theory of rings of the form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. 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 $K_*(R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2, (x_1,\ldots, x_d))$ when $R$ is a perfect $\mathbb{F}_p$-algebra. We conclude the paper with some other notable computations, including some rings which are not quite of the above form.