Tensor products and solutions to two homological conjectures for Ulrich modules

TL;DR

This paper investigates when tensor products of modules over Cohen-Macaulay rings are Ulrich, providing new criteria and applying these results to longstanding homological conjectures.

Contribution

It introduces conditions for tensor products to be Ulrich and proves the Auslander-Reiten and Huneke-Wiegand conjectures for Ulrich modules.

Findings

Tensor products are Ulrich under specific conditions.

Characterizations of complete intersections via Ulrich modules.

Homological conjectures hold for Ulrich modules.

Abstract

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we show that two celebrated homological conjectures, namely the Auslander-Reiten and the Huneke-Wiegand problems, are true for the class of Ulrich modules.