Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor

TL;DR

This paper investigates the vanishing properties of Ext and Tor for Cohen-Macaulay modules to characterize local rings and verify conjectures, extending previous results with new criteria based on finitely many vanishings.

Contribution

It introduces new test criteria for Cohen-Macaulay modules based on vanishing of Ext and Tor, and applies these to characterize local rings and prove the Auslander-Reiten Conjecture for modules of minimal multiplicity.

Findings

Finitely many Ext or Tor vanishings characterize certain Cohen-Macaulay modules.

Hypersurface rings of multiplicity at most two are characterized by these vanishing properties.

The Auslander-Reiten Conjecture is verified for all Cohen-Macaulay modules of minimal multiplicity.

Abstract

In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these test behaviour, along with other results, to characterize various kinds of local rings, including hypersurface rings of multiplicity at most two, surprisingly requiring only finitely many vanishing of Ext or Tor involving such CM modules. As further applications, we verify the long-standing (Generalized) Auslander-Reiten Conjecture for every CM module of minimal multiplicity over a Noetherian local ring, thus vastly extending a result of Huneke-\c{S}ega-Vraciu.