$K$-Theory of Cuspidal Curves Over a Perfectoid Base And Formal Analogues

TL;DR

This paper advances algebraic K-theory computations for cuspidal curves over perfectoid rings, extending previous characteristic p results to mixed characteristic and exploring related structures like the p-completed affine line.

Contribution

It extends K-theory calculations for cuspidal curves to mixed characteristic using perfectoid rings, building on recent advances and previous work.

Findings

K-theory of cuspidal curves over perfectoid rings is computed.

Results include the K-theory of p-completed cuspidal curves.

Analysis of the p-completed affine line over perfectoid rings.

Abstract

In this paper we continue the work of using the recent advances in algebraic $K$-theory to extend computations done in characteristic $p$ to the mixed characteristic setting using perfectoid rings. We extend the work of Hesselholt-Nikolaus in \cite{Hesselholt_Nikolaus} on the algebraic $K$-Theory of cuspidal curves. We consider both cuspidal curves and the $p$-completion of cuspidal curves. Along the way we also study the $K$-theory of the $p$-completed affine line over a perfectoid ring.