On Tor-vanishing of local rings

TL;DR

This paper investigates a variant of the Tor-vanishing property in local rings, showing that for generalized Golod rings, Tor-vanishing implies specific curvature conditions, and applies this to Gorenstein rings and the Auslander-Reiten conjecture.

Contribution

It establishes a new curvature-based criterion for Tor-vanishing in generalized Golod rings and applies it to Gorenstein rings, providing a unified approach to existing results.

Findings

Tor-vanishing implies curvature conditions in generalized Golod rings

Curvature of modules over these rings is restricted to {0, 1, curvature of residue field}

Applications to Gorenstein rings and the Auslander-Reiten conjecture

Abstract

Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the Tor-vanishing property if the converse holds, that is, $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $pd_R(M) < \infty$ or $pd_R(N) < \infty$. Interest in the Tor-vanishing property stems from the fact that Cohen-Macaulay local rings satisfying this property also satisfy the Auslander-Reiten conjecture. In this article, we study a variant of this property. If $R$ is a generalized Golod ring, we prove that $Tor^R_i(M, N) = 0$ for $i \gg 0$ implies $\{curv_R M, curv_R N \} \cap \{0, 1\} \neq \emptyset$. A key intermediate step in our proof is to show that $curv_R M \in \{0, 1, curv_R k\}$ for any module $M$ over a generalized Golod ring $R$. As an application, we prove that generic Gorenstein local rings, non-trivial connected sums of generalized Golod-Gorenstein rings satisfy the Tor-vanishing property and consequently the Auslander-Reiten conjecture. Our method suggests a uniform approach and recovers many old results on the Tor-vanishing property.