Progress on the Auslander-Reiten conjecture (Q2814342)

Let \(\Lambda\) be an Artinian (or a commutative Noetherian) ring and \(M\) a finitely generated \(\Lambda\)-module. Auslander-Reiten conjecture asserts that \(M\) is projective if \(\text{Ext}_\Lambda^i(M, M \oplus \Lambda) = 0\) for all \(i > 0\).NEWLINENEWLINEIn the present paper, authors proved a generalization of this conjecture for a special case. Let \(R\) be a commutative Gorenstein local ring of dimension \(d \geq 2\) and \(\Lambda\) an \(R\)-algebra. Assume that \(\Lambda\) is a finitely generated \(R\)-free module and that any \(R\)-projective \(\Lambda\)-module has finite projective dimension over \(\Lambda\). Let \(M\) be a finitely generated Gorenstein projective \(\Lambda\)-module such that \(M_{\mathfrak p}\) is \(\Lambda_{\mathfrak p}\)-projective for all non-maximal prime ideal \(\mathfrak p \subset R\). Then \(M\) is projective if \(\text{Ext}^{d-1}_\Lambda(M,M) = \text{Ext}^d_\Lambda(M,M) = 0\).