Gorenstein-projective and semi-Gorenstein-projective modules

Abstract

Let $A$ be an artin algebra. An $A$-module $M$ will be said to be semi-Gorenstein-projective provided that ${Ext}^i\left(M,A\right)=0$ for all $i\ge 1$. All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of semi-Gorenstein-projective modules which are not Gorenstein-projective have been known. One of the aims of the paper is to provide conditions on $A$ such that all semi-Gorenstein-projective left modules are Gorenstein-projective (we call such an algebra left weakly Gorenstein). In particular, we show that in case there are only finitely many isomorphism classes of indecomposable left modules which are both semi-Gorenstein-projective and torsionless, then $A$ is left weakly Gorenstein. On the other hand, we exhibit a 6-dimensional algebra $\mathrm{\Lambda }$ with a semi-Gorenstein-projective module $M$ which is not torsionless (thus not Gorenstein-projective). Actually, also the $\mathrm{\Lambda }$-dual module $M^{\ast }$ is semi-Gorenstein-projective. In this way, we show the independence of the total reflexivity conditions of Avramov and Martsinkovsky, thus completing a partial proof by Jorgensen and Şega. Since all the syzygy-modules of $M$ and $M^{\ast }$ are 3-dimensional, the example can be checked (and visualized) quite easily.