Gorenstein global dimensions relative to balanced pairs

Haiyu Liu; Rongmin Zhu; Yuxian Geng; Haiyu Liu; Rongmin Zhu; Yuxian Geng

doi:10.3934/era.2020082

Electronic Research Archive

2020, Volume 28, Issue 4: 1563-1571. doi: 10.3934/era.2020082

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Gorenstein global dimensions relative to balanced pairs

Haiyu Liu ^{1
,},
Rongmin Zhu ^{1
,
,},
Yuxian Geng ^{2
,}

1.
Department of Mathematics, Nanjing University, Nanjing 210093, China
2.
School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China

Received: 01 January 2020 Revised: 01 May 2020 Published: 31 July 2020
16E05, 16E30, 18E10, 18G20

Let $ \mathcal{G}(\mathcal{X}) $ and $ \mathcal{G}(\mathcal{Y}) $ be Gorenstein subcategories induced by an admissible balanced pair $ (\mathcal{X}, \mathcal{Y}) $ in an abelian category $ \mathcal{A} $. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of $ \mathcal{A} $ induced by the balanced pair $ (\mathcal{X}, \mathcal{Y}) $. As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring $ R $.
- Abelian category,
- balanced pair,
- Gorenstein subcategory,
- homological dimension,
- global dimension
Citation: Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs[J]. Electronic Research Archive, 2020, 28(4): 1563-1571. doi: 10.3934/era.2020082

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Abstract

Let $ \mathcal{G}(\mathcal{X}) $ and $ \mathcal{G}(\mathcal{Y}) $ be Gorenstein subcategories induced by an admissible balanced pair $ (\mathcal{X}, \mathcal{Y}) $ in an abelian category $ \mathcal{A} $. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of $ \mathcal{A} $ induced by the balanced pair $ (\mathcal{X}, \mathcal{Y}) $. As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring $ R $.

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