On modules with self Tor vanishing

Abstract

The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen, Nasseh, Sather-Wagstaff, and Şega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of the vanishing of Tor. This has led to the notion of Tor-persistent rings. Our main result shows that the class of Tor-persistent local rings is closed under a number of standard procedures in ring theory.

Keywords:

• G-dimension
• projective dimension
• Tor-persistent ring
• vanishing of Tor

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D05
• 13D07

Acknowledgments

We thank Avramov, Iyengar, Nasseh, and Sather-Wagstaff for useful comments and for making their article [5] available to us. Part of this work was completed when Holm visited West Virginia University in March 2018. He is grateful for the kind hospitality of the WVU Department of Mathematics.

Notes

1 Note that this work is announced under the different title Vanishing of endohomology over local rings in [6].