### Electronic Research Archive

2020, Issue 4: 1563-1571. doi: 10.3934/era.2020082
Special Issues

# Gorenstein global dimensions relative to balanced pairs

• 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 $R$.

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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• 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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