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
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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