Commutative Noetherian rings possessing a homological property
Let \(A\) be a commutative Noetherian local ring. The authors prove that \(\text{id}_A (M) \geq \text{pd}_A (M)\) for all finitely generated \(A\)-modules \(M\) if and only if either \(A\) is not a Cohen-Macaulay ring or \(A\) is a Gorenstein ring (where \(\text{id}\) and \(\text{pd}\) are used for the injective and projective dimensions). If \(A\) is a commutative Noetherian ring the set of all maximal ideals \(m\) of \(A\) for which \(A_m\) is a Cohen-Macaulay ring (respectively a Gorenstein ring) is denoted by \(\text{CM}(A)\) (respectively \(G(A))\). Then it is proved that \(\text{id}_A (M) \geq \text{pd}_A (M)\) for all finitely generated \(A\)-modules \(M\) if and only if \(\text{CM}(A) = G(A)\).
scientific article; zbMATH DE number 817782