[Paper Review] Direct limits of Gorenstein injective modules

Alina Iacob|arXiv (Cornell University)|Aug 16, 2023

Algebraic structures and combinatorial modelsMathematics3 citations

TL;DR

This paper resolves a long-standing open problem in Gorenstein homological algebra by proving that the class of Gorenstein injective modules is closed under arbitrary direct limits if and only if the ring is left noetherian and the character module of every Gorenstein injective module is Gorenstein flat. The equivalence is established through connections with duality pairs, cotorsion pairs, and properties of pure submodules and quotients.

ABSTRACT

One of the open problems in Gorenstein homological algebra is: when is the class of Gorenstein injective modules closed under arbitrary direct limits? It is known that if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under direct limits, then the ring is noetherian. The open problem is whether or not the converse holds. We give equivalent characterizations of $\mathcal{GI}$ being closed under direct limits. More precisely, we show that the following statements are equivalent:\\ (1) The class of Gorenstein injective left $R$-modules is closed under direct limits.\\ (2) The ring $R$ is left noetherian and the character module of every Gorenstein injective left $R$-module is Gorenstein flat.\\ (3) The class of Gorenstein injective modules is covering and it is closed under pure quotients.\\ (4) $\mathcal{GI}$ is closed under pure submodules.

Motivation & Objective

• To resolve the open problem of when the class of Gorenstein injective modules is closed under arbitrary direct limits.
• To determine whether the noetherian condition alone implies closure under direct limits, or whether additional conditions are required.
• To establish equivalent characterizations of the closure property using duality pairs, cotorsion pairs, and purity conditions.
• To investigate the relationship between the covering property of Gorenstein injective modules and their closure under direct limits.
• To explore sufficient conditions under which the character module of a Gorenstein injective module is Gorenstein flat.

Proposed method

• Use of duality pairs to relate Gorenstein injective and Gorenstein flat modules via character modules.
• Application of the complete hereditary cotorsion pair $ (^{ot}\mathcal{GI}, \mathcal{GI}) $ over any ring $ R $, as established in previous work.
• Leveraging the fact that direct limits are pure quotients of direct sums, linking closure under direct limits to closure under pure quotients.
• Employing the characterization that a class closed under pure submodules and pure quotients is closed under direct limits.
• Using the equivalence between the covering property and definability to deduce closure under pure submodules.
• Analyzing the injective dimension of modules and the flatness of character modules via dualizing complexes and Iwanaga-Gorenstein rings.

Experimental results

Research questions

• RQ1 Is the class of Gorenstein injective modules closed under arbitrary direct limits if and only if the ring is left noetherian and character modules of Gorenstein injective modules are Gorenstein flat?
• RQ2 Does every left noetherian ring satisfy the condition that the character module of every Gorenstein injective module is Gorenstein flat?
• RQ3 Is the covering property of the class $ \mathcal{GI} $ equivalent to its closure under direct limits?
• RQ4 Can the closure of $ \mathcal{GI} $ under direct limits be characterized via purity and duality pair structures?
• RQ5 What classes of rings (e.g., Gorenstein rings, rings with dualizing complexes) satisfy the condition that character modules of Gorenstein injective modules are Gorenstein flat?

Key findings

• The class $ \mathcal{GI} $ of Gorenstein injective modules is closed under arbitrary direct limits if and only if $ R $ is left noetherian and the character module of every Gorenstein injective left $ R $-module is Gorenstein flat.
• The class $ \mathcal{GI} $ is covering if and only if it is closed under pure quotients and pure submodules, which in turn implies closure under direct limits.
• If $ R $ is left noetherian and $ \operatorname{id}_R(R) \leq n $, then the character module of every Gorenstein injective module is Gorenstein flat.
• For commutative noetherian rings with a dualizing complex, the character modules of Gorenstein injective modules are Gorenstein flat.
• Iwanaga-Gorenstein rings satisfy the condition that character modules of Gorenstein injective modules are Gorenstein flat.
• The class $ \mathcal{GI} $ is closed under pure submodules and pure quotients if and only if it is closed under direct limits, under the assumption of the covering property.