Direct limits of Gorenstein injective modules
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2023
- DOI:
- 10.48550/arXiv.2308.08699
- arXiv:
- arXiv:2308.08699
- Bibcode:
- 2023arXiv230808699I
- Keywords:
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- Mathematics - Commutative Algebra;
- 16E05;
- 16E10