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arXiv · 0712.1328

When Are Torsionless Modules Projective?

Rong Luo·Zhaoyong Huang
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Abstract

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

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Rong Luo, Zhaoyong Huang. 2007-12-09. When Are Torsionless Modules Projective?. https://arxiv.org/abs/0712.1328

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