On modules with self Tor vanishing

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 \c{S}ega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of 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.Comment: Introduction has been rewritten and terminology has been changed to align with work of Avramov, Iyengar, Nasseh, and Sather-Wagstaff. 5 page

Equivalences from tilting theory and commutative algebra from the adjoint functor point of view

We give a category theoretic approach to several known equivalences from (classic) tilting theory and commutative algebra. Furthermore, we apply our main results to establish a duality theory for relative Cohen-Macaulay modules in the sense of Hellus, Schenzel, and Zargar.Comment: This is the final version (17 pages) to appear in the New York Journal of Mathematic