Direct limits of Gorenstein injective modules (2308.08699v1)

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.