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

Abstract

In this paper we continue the work, started in [Rig20], of using the recent advances in algebraic Ktheory to extend computations done in characteristic p to the mixed characteristic setting using perfectoid rings. We extend the work of Hesselholt-Nikolaus in [HN20] 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.