When Are Torsionless Modules Projective?

TL;DR

This paper characterizes when finitely generated torsionless modules over Artinian rings are projective, providing specific Ext vanishing conditions that guarantee projectivity, especially in the Gorenstein case.

Contribution

It establishes new criteria involving Ext vanishing for torsionless modules to be projective over Artinian rings, extending previous understanding.

Findings

Torsionless modules over Artinian local algebras with radical square zero are projective if Ext^1 vanishes.

Over commutative Artinian rings, certain Ext conditions imply torsionless modules are projective.

Gorenstein projective modules are projective iff they are selforthogonal under these conditions.

Abstract

In this paper, we study the problem when a finitely generated torsionless module is projective. Let $\Lambda$ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless $\Lambda$-module $M$ is projective if ${\rm Ext^1_\Lambda}(M,M)=0$. For a commutative Artinian ring $\Lambda$, a finitely generated torsionless $\Lambda$-module $M$ is projective if the following conditions are satisfied: (1) ${\rm Ext}^i_{\Lambda}(M,\Lambda)=0$ for $i=1,2,3$; and (2) ${\rm Ext}^i_{\Lambda}(M,M)=0$ for $i=1,2$. As a consequence of this result, we have that for a commutative Artinian ring $\Lambda$, a finitely generated Gorenstein projective $\Lambda$-module is projective if and only if it is selforthogonal.