Cartan-Eilenberg Gorenstein-injective m-complexes

• Main Authors: Bo Lu, Angmao Daiqing
• Format: Article
• Language: English
• Published: AIMS Press 2021-02-01
• Series: AIMS Mathematics
• Subjects: gorenstein-injective module; cartan-eilenberg gorenstein-injective m-complex; two-degree cartan-eilenberg gorenstein-injective m-complex; stability
• Online Access: http://www.aimspress.com/article/doi/10.3934/math.2021255?viewType=HTML

We study the notion of Cartan-Eilenberg Gorenstein-injective $m$-complexes. We show that a $m$-complex $G$ is Cartan-Eilenberg Gorenstein-injective if and only if $G_n$, $\mathrm{Z}_n^{t}(G)$, $\mathrm{B}_n^{t}(G)$ and $\mathrm{H}_n^{t}(G)$ are Gorenstein-injective modules for each $n\in\mathbb{Z}$ and $t=1,2,\ldots,m$. As an application, we show that an iteration of the procedure used to define the Cartan-Eilenberg Gorenstein-injective $m$-complexes yields exactly the Cartan-Eilenberg Gorenstein-injective $m$-complexes. Specifically, given a Cartan-Eilenberg exact sequence of Cartan-Eilenberg Gorenstein-injective $m$-complexes $$\mathbb{G}=\cdots\rightarrow G^{-1}\rightarrow G^0\rightarrow G^1\rightarrow \cdots$$ such that the functor $\mathrm{Hom}_{\mathcal{C}_m({R})}(H,-)$ leave $\mathbb{G}$ exact for each Cartan-Eilenberg Gorenstein-injective $m$-complex $H$, then $\mathrm{Ker}(G^0\rightarrow G^1)$ is a Cartan-Eilenberg Gorenstein-injective $m$-complex.