$K$-Theory of Truncated Polynomials

TL;DR

This paper investigates the algebraic K-theory of truncated polynomial rings using trace methods, filtrations, and prismatic cohomology, providing explicit computations for various classes of rings including perfectoid rings and discrete valuation rings.

Contribution

It introduces new computational techniques for algebraic K-theory of truncated polynomial rings via trace methods and prismatic cohomology, extending previous results to broader classes of rings.

Findings

Computed K-theory for perfectoid rings using big Witt vectors

Derived K-theory for smooth curves over perfectoid rings via prismatic cohomology

Analyzed K-theory of discrete valuation rings with perfect residue fields through prismatic cohomology and Hodge-Tate divisor

Abstract

We study the algebraic $K$-theory of rings of the form $R[x]/x^e$. We do this via trace methods and filtrations on topological Hochschild homology and related theories by quasisyntomic sheaves. We produce computations for $R$ a perfectoid ring in terms of the big Witt vectors of $R$, for $R$ a smooth curve over a perfectoid ring in terms of the prismatic cohomology of $R$, and for $R$ a complete mixed characteristic discrete valuation rings with perfect residue field in terms of the prismatic cohomology and Hodge-Tate divisor of $R$.