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3. Duality Pairs Induced by One-Sided Gorenstein Subcategories

Author

Song, Weiling, Zhao, Tiwei, and Huang, Zhaoyong

Subjects
Mathematics - Category Theory, Mathematics - Rings and Algebras, 18G25, 16E30
Abstract

For a ring $R$ and an additive subcategory $\C$ of the category $\Mod R$ of left $R$-modules, under some conditions we prove that the right Gorenstein subcategory of $\Mod R$ and the left Gorenstein subcategory of $\Mod R^{op}$ relative to $\C$ form a coproduct-closed duality pair. Let $R,S$ be rings and $C$ a semidualizing ($R,S$)-bimodule. As applications of the above result, we get that if $S$ is right coherent and $C$ is faithfully semidualizing, then $(\mathcal{GF}_C(R),\mathcal{GI}_C(R^{op}))$ is a coproduct-closed duality pair and $\mathcal{GF}_C(R)$ is covering in $\Mod R$, where $\mathcal{G}\mathcal{F}_C(R)$ is the subcategory of $\Mod R$ consisting of $C$-Gorenstein flat modules and $\mathcal{G}\mathcal{I}_C(R^{op})$ is the subcategory of $\Mod R^{op}$ consisting of $C$-Gorenstein injective modules; we also get that if $S$ is right coherent, then $(\mathcal{A}_C(R^{op}),l\mathcal{G}(\mathcal{F}_C(R)))$ is a coproduct-closed and product-closed duality pair and $\mathcal{A}_C(R^{op})$ is covering and preenveloping in $\Mod R^{op}$, where $\mathcal{A}_C(R^{op})$ is the Auslander class in $\Mod R^{op}$ and $l\mathcal{G}(\mathcal{F}_C(R))$ is the left Gorenstein subcategory of $\Mod R$ relative to $C$-flat modules.

Published
2020
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4. One-Sided Gorenstein Subcategories

Author

Song, Weiling, Zhao, Tiwei, and Huang, Zhaoyong

Subjects
Mathematics - Category Theory, Mathematics - Rings and Algebras, 18G25, 16E10, 18G10
Abstract

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\C$ of an abelian category $\A$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\C$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\A$ with finite $r\mathcal{G}(\C)$-projective dimension is isomorphic to a kernel (resp. a cokernel) of a morphism from an object in $\A$ with finite $\C$-projective dimension to an object in $r\mathcal{G}(\C)$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\A$ having enough injectives.

Published
2020
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21. Multifunctional Zeolitic Imidazolate Framework Nanocomposites as Fenton Catalysts for Synergistic Chemodynamic-Photothermal-Chemo Therapy

Author

Song, Weiling, primary, Yang, Huan, additional, Sun, Yujie, additional, Yang, Yuanming, additional, Dong, Qi, additional, He, Peng, additional, Liu, Gengjun, additional, Zhao, Zhen, additional, Qin, Lifeng, additional, Gao, Shunxiang, additional, Liu, Jing, additional, Zhou, Hong, additional, and Wang, Haiyan, additional

Published
2023
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23. Polyhedral Au Nanoparticle/MoOx Heterojunction-Enhanced Ultrasensitive Dual-Mode Biosensor for miRNA Detection Combined with a Nonenzymatic Cascade DNA Amplification Circuit

Author

Zhao, Lin, primary, Li, Tiantian, additional, Xu, Xinlin, additional, Xu, Yang, additional, Li, Dongxiang, additional, Song, Weiling, additional, Zhan, Tianrong, additional, He, Peng, additional, Zhou, Hong, additional, Xu, Jing-Juan, additional, and Chen, Hong-Yuan, additional

Published
2023
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42. THE SOCLE OF THE LAST TERM IN THE MINIMAL INJECTIVE RESOLUTION OF A GORENSTEIN MODULE

Author

Song, Weiling, Zhang, Xiaojin, and Huang, Zhaoyong

Subjects
Mathematics::Commutative Algebra, 16E05, Mathematics::Rings and Algebras, 16E10, 16E30
Abstract

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_RU$ a Gorenstein module with $S={\rm End}(_RU)$. If the injective dimensions of $_RU$ and $U_S$ are finite, then the last term in the minimal injective resolution of $_{R}U$ has an essential socle.

Published
2019

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