On Modules M such that both M and M ∗ are Semi-Gorenstein-Projective

Let A be an artin algebra. An A-module M is semi-Gorenstein-projective provided that Exti(M,A) = 0 for all i ≥ 1. If M is Gorenstein-projective, then both M and its A-dual M∗ are semi-Gorenstein projective. As we have shown recently, the converse is not true, thus answering a question raised by Avramov and Martsinkovsky. The aim of the present note is to analyse in detail the modules M such that both M and M∗ are semi-Gorenstein-projective.