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A twisted Bass-Heller-Swan decomposition for localising invariants

Dominik Kirstein, Christian Kremer

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.

A twisted Bass-Heller-Swan decomposition for localising invariants

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.

Paper Structure

This paper contains 12 sections, 28 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Recollections on K-theory of pushouts
  3. The main theorem
  4. Twisted endomorphisms and automorphisms
  5. Twisted nilpotent endomorphisms
  6. K-theory of twisted automorphisms
  7. K-theory of twisted endomorphisms
  8. Regularity
  9. Applications and Examples
  10. K-theory of tensor algebras
  11. K-theory of mapping tori
  12. A-theoretic Nil-terms

Key Result

Theorem 1

Let $\alpha \colon \mathcal{C} \to \mathcal{C}$ be an exact endofunctor of an idempotent complete stable $\infty$-category and $E \colon \mathrm{Cat}^{\mathrm{perf}} \to \mathcal{E}$ a localising invariant. The assembly map $E(\mathcal{C})_{h \mathbb{N}} \to E(\mathcal{C}_{h \mathbb{N}})$ has a natu

Theorems & Definitions (54)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 2.6: LandTamme23
  • Definition 3.2: Twisted endormophisms and automorphisms
  • Proposition 3.6
  • proof
  • Lemma 3.7
  • ...and 44 more