When Are Torsionless Modules Projective?

In this paper, we study the problem when a finitely generated torsionless module is projective. Let $\mathrm{\Lambda }$ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless $\mathrm{\Lambda }$-module $M$ is projective if $\mathrm{E}\mathrm{x}{\mathrm{t}}_{\mathrm{\Lambda }}^1(M,M)=0$. For a commutative Artinian ring $\mathrm{\Lambda }$, a finitely generated torsionless $\mathrm{\Lambda }$-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 $\mathrm{\Lambda }$, a finitely generated Gorenstein projective $\mathrm{\Lambda }$-module is projective if and only if it is selforthogonal.