Maximal Cohen-Macaulay tensor products and vanishing of Ext modules

TL;DR

This paper studies when tensor products of modules are maximal Cohen-Macaulay and provides criteria for module projectivity based on Ext vanishing, with applications to the Auslander-Reiten conjecture in Cohen-Macaulay rings.

Contribution

It introduces new criteria linking tensor products and Ext vanishing to module properties, and proves the Auslander-Reiten conjecture for Cohen-Macaulay normal rings.

Findings

Tensor products are maximal Cohen-Macaulay under specific conditions.

Projectivity of modules can be characterized by Ext vanishing.

The Auslander-Reiten conjecture holds for Cohen-Macaulay normal rings.

Abstract

In this paper, we investigate the maximal Cohen-Macaulay property of tensor products of modules, and then give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the Auslander-Reiten conjecture holds for Cohen-Macaulay normal rings.