Verschiebung maps among $K$-groups of truncated polynomial algebras

TL;DR

This paper investigates Verschiebung maps among K-groups of truncated polynomial algebras over rings with nilpotent prime p, providing explicit evaluations using topological Hochschild homology and de Rham-Witt forms, with applications to perfectoid fields.

Contribution

It offers explicit calculations of Verschiebung maps in K-theory for truncated polynomial algebras, connecting algebraic K-theory with topological Hochschild homology and de Rham-Witt groups, and applies results to perfectoid fields.

Findings

Evaluated Verschiebung maps in K-theory for general rings with nilpotent p.

Expressed these maps in terms of topological Hochschild homology.

Calculated relative K-groups for certain perfectoid fields.

Abstract

Let $p$ be a prime number, and let $A$ be a ring in which $p$ is nilpotent. In this paper, we consider the maps $$K_{q+1}(A[x]/(x^m), (x))\to K_{q+1}(A[x]/(x^{mn}), (x)),$$induced by the ring homomorphism $A[x]/(x^{m})\to A[x]/(x^{mn})$, $x\mapsto x^n$. We evaluate these maps, up to extension, for general $A$ in terms of topological Hochschild homology, and for regular $\mathbb{F}_p$-algebras $A$, in terms of groups of de Rham-Witt forms. After the evaluation, we give a calculation of the relative $K$-group of $\mathcal{O}_{K}/p\mathcal{O}_{K}$ for certain perfectoid fields $K$.