Let A be an artin algebra. An A-module M will be called a semi-Gorenstein-projective module provided that $Ext^i(M,A) = 0$ 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 $\Lambda$ with a semi-Gorenstein-projective module M which is not torsionless (thus not Gorenstein-projective). Actually, also the $\Lambda$-dual module $M^*$ is semi-Gorenstein-projective.  In this way, we complete the solution to the open problem showing the independence of the total reflexivity conditions of Avramov and Martsinkovsky, which started by Jorgensen and Sega.
This is a joint work with Claus Michael Ringel.