A criterion for dualizing modules

TL;DR

This paper provides a new criterion to identify dualizing modules among semidualizing modules over finite dimensional commutative Noetherian rings, based on the behavior of Tor functors.

Contribution

It establishes a characterization of dualizing modules using the C-injectivity of Tor modules between C-injective modules, offering a new perspective in module theory.

Findings

C is dualizing iff Tor_i^R(E,E') is C-injective for all C-injective E, E' and i≥0

Provides a practical criterion for identifying dualizing modules

Enhances understanding of the structure of semidualizing modules

Abstract

We establish a characterization of dualizing modules among semidualizing modules. Let R be a finite dimensional commutative Noetherian ring with identity and C a semidualizing R-module. We show that C is a dualizing R-module if and only if Tor_i^R(E,E') is C- injective for all C-injective R-modules E and E' and all i\geq 0.