# Extremal growth of Betti numbers and trivial vanishing of (co)homology

## TL;DR

This paper investigates conditions under which trivial vanishing occurs in Cohen-Macaulay local rings, linking it to the growth of Betti and Bass numbers, and applies these results to settle cases of the Auslander-Reiten conjecture and criteria for Gorenstein rings.

## Contribution

It establishes obstructions to trivial vanishing via Betti and Bass number growth and provides new criteria for Gorenstein properties and the Auslander-Reiten conjecture.

## Key findings

- Obstructions for failure of trivial vanishing based on Betti and Bass numbers.
- Sufficient conditions for trivial vanishing in generalized Golod rings.
- Criteria for Gorenstein property from vanishing Ext sequences.

## Abstract

A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ for all large $i$ implies $M$ has finite projective dimension or $N$ has finite injective dimension. In this paper, we establish obstructions for the failure of trivial vanishing in terms of the asymptotic growth of the Betti and Bass numbers of the modules involved. These, together with a result of Gasharov and Peeva, provide sufficient conditions for $R$ to satisfy trivial vanishing; we provide sharpened conditions when $R$ is generalized Golod. Our methods allow us to settle the Auslander-Reiten conjecture in several new cases. In the last part of the paper, we provide criteria for the Gorenstein property based on consecutive vanishing of Ext. The latter results improve similar statements due to Ulrich, Hanes-Huneke, and Jorgensen-Leuschke.