On the Gorenstein projective conjecture: IG-projective modules (Q2803572)

A finitely generated module \(M\) over a commutative Noetherian local ring \(R\) is \textit{Gorenstein projective} if it is isomorphic to the image of \(P_0\rightarrow P_{-1}\), where \(P_i\rightarrow P_{i-1}\) is a bi-infinite exact sequence of projective modules such that the hom-functor \(\mathrm{Hom}_R(-,R)\) is exact. Although a projective module is Gorenstein projective, the converse is not necessarily true. A Gorenstein projective \(R\)-module is said to be \textit{IG-projective} if it is the direct sum of indecomposable Gorenstein projective modules \(N\) that admit either an irreducible epimorphism \(P\rightarrow N\) or and irreducible monomorphism \(N\rightarrow P\), with \(P\) a projective module. The authors show that an IG-projective module \(M\) is projective if and only if it is self-orthogonal; that is, \(\mathrm{Ext}_R^i(M,M)\) is trivial for all \(i>0\).