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Asymptotic vanishing of cohomology in triangulated categories

Petter Andreas Bergh, David A. Jorgensen, Peder Thompson

TL;DR

This work develops a unified framework for the asymptotic vanishing of cohomology in triangulated categories with a central ring action. Under an eventual Noetherian condition on $\text{Hom}_{\mathscr{C}}^{\ge n_0}(A,B)$, the main theorem yields a sharp dichotomy: either cohomology groups vanish beyond some threshold, or they persist nontrivially along a single residue class modulo a period $d$, with $d$ determined by the generating degrees of the even part of the relevant ring quotient. The authors apply this result to commutative local complete intersections, finite-dimensional algebras (including quantum complete intersections and exterior algebras), and modular group algebras, obtaining concrete residue-class patterns for $\operatorname{Ext}$-groups. These findings unify and extend classical asymptotic vanishing phenomena, offering practical criteria for predicting long-term behavior of cohomology in representation theory and homological algebra.

Abstract

Given a graded-commutative ring acting centrally on a triangulated category, our main result shows that if cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.

Asymptotic vanishing of cohomology in triangulated categories

TL;DR

This work develops a unified framework for the asymptotic vanishing of cohomology in triangulated categories with a central ring action. Under an eventual Noetherian condition on , the main theorem yields a sharp dichotomy: either cohomology groups vanish beyond some threshold, or they persist nontrivially along a single residue class modulo a period , with determined by the generating degrees of the even part of the relevant ring quotient. The authors apply this result to commutative local complete intersections, finite-dimensional algebras (including quantum complete intersections and exterior algebras), and modular group algebras, obtaining concrete residue-class patterns for -groups. These findings unify and extend classical asymptotic vanishing phenomena, offering practical criteria for predicting long-term behavior of cohomology in representation theory and homological algebra.

Abstract

Given a graded-commutative ring acting centrally on a triangulated category, our main result shows that if cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
Paper Structure (3 sections, 9 theorems, 15 equations)

This paper contains 3 sections, 9 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Applications

Key Result

Theorem 1

Let $( \mathscr{C}, \Sigma, \Delta )$ be a triangulated category with a central ring action from a graded-commutative non-negatively graded ring $H$, and $A,B \in \mathscr{C}$ two objects such that the $H$-module $\operatorname{Hom}_{\mathscr{C}}^{\ge n_0}(A,B)$ is Noetherian for some $n_0 \in \math Then either $\operatorname{Hom}_{\mathscr{C}}(A, \Sigma^n B) = 0$ for all $n\gg 0$, or there exist

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Theorem 3.1
  • ...and 11 more