Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor
Souvik Dey, Dipankar Ghosh, Aniruddha Saha
TL;DR
This work develops test properties for CM modules through finitely many vanishing of Ext or Tor, focusing on modules with $e(M) < 2\mu(M)$ or minimal multiplicity, to detect finiteness of projective or injective dimensions of other modules. It yields broad characterizations of regular, hypersurface, and complete intersection local rings via vanishing criteria and exhibits that modules of minimal multiplicity satisfy the (Generalized) Auslander–Reiten Conjecture over any Noetherian local ring. The approach blends multiplicity theory, complexity, and canonical-module dualities to translate homological vanishing into structural ring properties, with significant implications for Gorensteinness and AR-type conjectures. Overall, the paper provides finite-vanishing criteria that connect CM-module invariants to global ring-theoretic regularity and singularity types, advancing understanding of how Ext/Tor vanishings control homological dimensions and module freeness.
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-Şega-Vraciu.