Canonical Filtrations of Gorenstein Injective Modules

TL;DR

This paper demonstrates that Gorenstein injective modules over Gorenstein rings of finite Krull dimension have canonical filtrations, enabling the proof that their tensor products are also Gorenstein injective, supporting the homological algebra analogy.

Contribution

It introduces canonical filtrations for Gorenstein injective modules over Gorenstein rings of finite Krull dimension, establishing tensor product closure.

Findings

Gorenstein injective modules admit filtrations similar to indecomposable decompositions.

Tensor products of Gorenstein injective modules over these rings are Gorenstein injective.

Supports the homological algebra principle in Gorenstein contexts.

Abstract

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the principle. Over commutative, noetherian rings injective modules have direct sum decompositions into indecomposable modules. We will show that Gorenstein injective modules over Gorenstein rings of finite Krull dimension have filtrations analogous to those provided by these decompositions. This result will then provide us with the tools to prove that all tensor products of Gorenstein injective modules over these rings are Gorenstein injective.