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arXiv · 2308.08699

Direct limits of Gorenstein injective modules

Alina Iacob
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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.

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Alina Iacob. 2023-08-16. Direct limits of Gorenstein injective modules. https://arxiv.org/abs/2308.08699

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