Presentations of rings with non-trivial semidualizing modules

TL;DR

This paper characterizes Cohen-Macaulay rings with non-trivial semidualizing modules as certain homomorphic images of Gorenstein rings with decomposable defining ideals, expanding classical dualizing module results.

Contribution

It provides a new characterization of Cohen-Macaulay rings with non-trivial semidualizing modules via their relation to Gorenstein rings and ideal decompositions.

Findings

Cohen-Macaulay rings with non-trivial semidualizing modules are homomorphic images of Gorenstein rings.

Such rings have defining ideals that decompose cohomologically independently.

The result generalizes classical dualizing module characterizations.

Abstract

Let R be a commutative noetherian local ring. A finitely generated R-module C is semidualizing if it is self-orthogonal and satisfies the condition 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.