Presentations of rings with non-trivial semidualizing modules

Abstract

Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and \({{\rm Hom}_R(C,C)\cong R}\) . We prove that a Cohen–Macaulay ring R with dualizing module D admits a semidualizing module C satisfying \({R\ncong C \ncong D}\) if and only if it is a homomorphic image of a Gorenstein ring in which the defining ideal decomposes in a cohomologically independent way. This expands on a well-known result of Foxby, Reiten and Sharp saying that R admits a dualizing module if and only if R is Cohen–Macaulay and a homomorphic image of a local Gorenstein ring.