Asymptotic vanishing of cohomology in triangulated categories

TL;DR

This paper investigates the behavior of cohomology in triangulated categories, demonstrating that finite generation over a central ring leads to predictable asymptotic vanishing patterns, with broad applications.

Contribution

It establishes a link between finite generation of cohomology and asymptotic vanishing in triangulated categories, providing new insights and applications.

Findings

Finite generation over a central ring implies well-behaved asymptotic vanishing.

Enough consecutive vanishing leads to eventual vanishing of cohomology.

Several key applications demonstrate the theory's utility.

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.