Vanishing of Tor Over Complete Intersections

In this paper we are concerned with the vanishing of $\textnormal{Tor}$ over complete intersection rings. Building on results of C. Huneke, D. Jorgensen and R. Wiegand, and, more recently, H. Dao, we obtain new results showing that good depth properties on the $R$-modules $M$, $N$ and $M\otimes_RN$ force the vanishing of $\textnormal{Tor}^{R}_{i}(M,N)$ for all $i\geq 1$.Comment: 25 pages, to appear in the Journal of Commutative Algebr

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