The balance of relative Ext groups defined by semi dualizing modules

TL;DR

This paper investigates conditions under which certain relative Ext bifunctors coincide in the context of commutative Noetherian rings with semidualizing modules, establishing that this occurs precisely when the module is projective.

Contribution

It characterizes when the Ext bifunctors induced by $C$-projective and $C$-injective modules coincide, showing this is equivalent to $C$ being a projective module, and provides new criteria for projectivity.

Findings

Ext bifunctors coincide iff $C$ is projective.

Provides criteria for $C$ to be projective using cotorsion theories.

Establishes equivalence conditions for semidualizing modules.

Abstract

Let $R$ be a commutative Noetherian ring with identity and $C$ a semidualizing module for $R$. Let $\mathscr{P}_C(R)$ and $\mathscr{I}_C (R)$ denote, respectively, the classes of $C$-projective and $C$-injective $R$-modules. We show that their induced Ext bifunctors $\text{Ext}^i_{\mathscr{P}_C}(-,\sim)$ and $\text{Ext}^i_{\mathscr{I}_C}(-,\sim)$ coincide for all $i\geq 0$ if and only if $C$ is projective. Also, we provide some other criteria for $C$ to be projective by using some special cotorsion theories.