A twisted Bass-Heller-Swan decomposition for localising invariants

TL;DR

This paper extends the Bass-Heller-Swan decomposition to localising invariants of categories with twisted automorphisms, providing new splitting formulas and identifying Nil-terms in algebraic K-theory.

Contribution

It introduces a generalized splitting for localising invariants in twisted settings and characterizes Nil-terms as twisted endomorphisms and nilpotent endomorphisms.

Findings

Splitting formulas for Waldhausen's A-theory of mapping tori.

Identification of Nil-terms as twisted endomorphisms.

Generalization of vanishing results for Nil-terms in regular rings.

Abstract

We generalise the classical Bass-Heller-Swan decomposition for the K-theory of (twisted) Laurent algebras to a splitting for general localising invariants of certain categories of twisted automorphisms. As an application, we obtain splitting formulas for Waldhausen's A-theory of mapping tori and for the K-theory of certain tensor algebras. We identify the Nil-terms appearing in this splitting in two ways. Firstly, as the reduced K-theory of twisted endomorphisms. Secondly, as the reduced K-theory of twisted nilpotent endomorphisms. Finally, we generalise classical vanishing results for Nil-terms of regular rings to our setting.