When Are Torsionless Modules Projective?

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 $\mathrm{Ex}{\mathrm{t}}_{\mathrm{\Lambda }}^1(M,M)=0$. For a commutative Artinian ring $\Lambda$, a finitely generated torsionless $\Lambda$-module $M$ is projective if the following conditions are satisfied: (1) ${\mathrm{Ext}}_{\Lambda }^i(M,\Lambda )=0$ for $i=1,2,3$; and (2) ${\mathrm{Ext}}_{\Lambda }^i(M,M)=0$ 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.

10 pages