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Vanishing of Nil-terms and negative K-theory for additive categories

Arthur Bartels, Wolfgang Lueck

TL;DR

The paper extends regular coherence from rings to additive categories by embedding into the abelian category of ${\mathbb{Z}\mathcal{A}}$-modules and develops intrinsic intrinsic regularity notions that yield vanishing results for twisted Nil-terms and negative $K$-groups. It then analyzes nested sequences of additive categories via sequence and limit categories ${\mathcal{S}}(\mathcal{A}_*)$ and ${\mathcal{L}}(\mathcal{A}_*)$, establishing uniform regular coherence to control their algebraic $K$-theory. A Bass-Heller-Swan framework for both connective and non-connective $K$-theory is extended to this setting, including twisted Laurent categories and projective-line constructions, enabling a main assembly theorem for $K^{\infty}$ with $\mathbb{Z}^r$-actions. The results provide structural vanishing and decomposition results essential for computing the algebraic $K$-theory of nested additive categories and have applications to Hecke algebras of reductive $p$-adic groups, contributing to Farrell-Jones style reductions in this context.

Abstract

We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance the negative K-groups and all twisted Nil-groups vanish for an additive category if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.

Vanishing of Nil-terms and negative K-theory for additive categories

TL;DR

The paper extends regular coherence from rings to additive categories by embedding into the abelian category of -modules and develops intrinsic intrinsic regularity notions that yield vanishing results for twisted Nil-terms and negative -groups. It then analyzes nested sequences of additive categories via sequence and limit categories and , establishing uniform regular coherence to control their algebraic -theory. A Bass-Heller-Swan framework for both connective and non-connective -theory is extended to this setting, including twisted Laurent categories and projective-line constructions, enabling a main assembly theorem for with -actions. The results provide structural vanishing and decomposition results essential for computing the algebraic -theory of nested additive categories and have applications to Hecke algebras of reductive -adic groups, contributing to Farrell-Jones style reductions in this context.

Abstract

We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance the negative K-groups and all twisted Nil-groups vanish for an additive category if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.

Paper Structure

This paper contains 46 sections, 60 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. $\mathbb{Z}$-categories, additive categories, and idempotent completions
  3. $\mathbb{Z}$-categories
  4. Additive categories
  5. Idempotent completion
  6. Twisted finite Laurent category
  7. The algebraic $K$-theory of $\mathbb{Z}$-categories
  8. The Bass-Heller-Swan decomposition for additive categories
  9. $\mathbb{Z}\mathcal{A}$-modules and the Yoneda embedding
  10. Basics about $\mathbb{Z}\mathcal{A}$-modules
  11. The Yoneda embedding
  12. Regularity properties of additive categories
  13. Definition of regularity properties in terms of the Yoneda embedding
  14. The definitions of the regularity properties for rings and additive categories are compatible
  15. Intrinsic definitions of the regularity properties
  16. ...and 31 more sections

Key Result

Lemma 3.2

Let $I \colon \mathcal{A} \to \mathcal{A}'$ be the inclusion of a full cofinal additive subcategory.

Theorems & Definitions (130)

  • Definition 2.1: Twisted finite Laurent category $\mathcal{A}_{\Phi}[t,t^{-1}]$
  • Example 2.2
  • Definition 2.3: $\mathcal{A}_{\Phi}[t]$ and $\mathcal{A}_{\Phi}[t^{-1}]$
  • Definition 3.1: Algebraic $K$-theory of $\mathbb{Z}$-categories
  • Lemma 3.2
  • proof
  • Theorem 4.1: The Bass-Heller-Swan decomposition for non-connective $K$-theory of additive categories
  • Theorem 4.2: Fundamental sequence of $K$-groups
  • proof
  • Theorem 4.3: The Bass-Heller-Swan decomposition for connective $K$-theory of additive categories
  • ...and 120 more