# On the K-theory of truncated polynomial algebras, revisited

## TL;DR

This paper revisits the computation of K-theory for truncated polynomial algebras over perfect fields of positive characteristic, simplifying previous methods by using the Nikolaus-Scholze framework and homology analysis.

## Contribution

It provides a new proof of the K-theory of truncated polynomial algebras using modern homotopy techniques, avoiding complex geometric arguments.

## Key findings

- K-groups expressed via big Witt vectors
- Simplified proof using Nikolaus-Scholze framework
- Results applicable to perfect fields of positive characteristic

## Abstract

We revisit the computation, due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras for perfect fields of positive characteristic. The resulting K-groups are expressed in terms of big Witt vectors of the field. The original proof relied on an understanding of cyclic polytopes in order to determine the genuine equivariant homotopy type of the cyclic bar construction for a suitable monoid. Using the Nikolaus-Scholze framework for topological cyclic homology we achieve the same result using only the homology of said cyclic bar construction, as well as the action of Connes' operator.