Gorenstein injective filtrations over Cohen-Macaulay rings with dualizing modules

TL;DR

This paper extends the theory of Gorenstein injective modules over Cohen-Macaulay rings with dualizing modules, showing they admit filtrations similar to classical injective modules, and explores their properties and limitations.

Contribution

It introduces Gorenstein injective filtrations over Cohen-Macaulay rings with dualizing modules, extending classical results and analyzing their properties and limitations.

Findings

Gorenstein injective modules admit filtrations over Cohen-Macaulay rings with dualizing modules.

Gorenstein injective modules are not closed under tensor or torsion products in certain rings.

Filtrations do not necessarily produce direct sum decompositions in these contexts.

Abstract

Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective $R$-modules need not be closed under tensor products, even when $R$ is local and artinian; (2) the class of Gorenstein injective $R$-modules need not be closed under torsion products, even when $R$ is a local, complete hypersurface; and (3) the filtrations given in our main theorem do not yield direct sum decompositions, even when $R$ is a local, complete hypersurface.