Cartan-Eilenberg Gorenstein-injective $ m $-complexes

Bo Lu; Angmao Daiqing; Bo Lu; Angmao Daiqing

doi:10.3934/math.2021255

AIMS Mathematics

2021, Volume 6, Issue 5: 4306-4318. doi: 10.3934/math.2021255

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Research article

Cartan-Eilenberg Gorenstein-injective $ m $-complexes

Bo Lu ^,,
Angmao Daiqing

College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, Gansu, P. R. China

Received: 30 October 2020 Accepted: 01 February 2021 Published: 08 February 2021
MSC : 18E10, 18G25, 18G35

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.
Citation: Bo Lu, Angmao Daiqing. Cartan-Eilenberg Gorenstein-injective $ m $-complexes[J]. AIMS Mathematics, 2021, 6(5): 4306-4318. doi: 10.3934/math.2021255

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Abstract

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.

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