Abelian范畴中n-挠对的粘合

Recollements of n-Torsion Pairs in Abelian Categories

摘要

首先,定义并研究 Abelian范畴中的n-挠对,并设($\mathscr{A}, \mathscr{B}, \mathscr{C}$)是 Abelian范畴中的粘合. 其次,利用高阶扩张和推出拉回等方法证明: $\mathscr{A}$ 和 $\mathscr{C}$ 中的n-挠对可诱导 $\mathscr{B}$ 中的n-挠对;反之, $\mathscr{B}$中的n-挠对也可诱导 $\mathscr{A}$和$\mathscr{C}$中的n-挠对.

Abstract

Firstly, we define and study n-torsion pairs in Abelian categories, and let (($\mathscr{A}, \mathscr{B}, \mathscr{C}$) be a recollement of Abelian categories. Secondly, by using methods such as higher extensions, pushout, and pullback, we prove that an n-torsion pair in $\mathscr{A}$ and $\mathscr{C}$ can induce an n-torsion pair in $\mathscr{B}$. Conversely, an n-torsion pair in $\mathscr{B}$ can also induce an n-torsion pair in $\mathscr{A}$ and $\mathscr{C}$.

关键词

Abelian范畴 / n-挠对 / 粘合

Key words

Abelian category / n-torsion pair / recollement

参考文献

[1] Dickson S E. A Torsion Theory for Abelian Categories[J]. Transactions of the American Mathematical Society, 1966, 121: 223-235.

[2] Fan C Y, Yao H L. Torsion Pairs in Triangulated Categories[J]. Advances in Pure Mathematics, 2013, 3(3): 374-379.

[3] Tattar A. Torsion Pairs and Quasi-Abelian Categories[J]. Algebras and Representation Theory, 2020, 24(6): 1557-1581.

[4] He J, Hu Y G, Zhou P Y. Torsion Pairs and Recollements of Extriangulated Categories[J]. Communications in Algebra, 2022, 50(5): 2018-2036.

[5] Ma X, Huang Z Y. Torsion Pairs in Recollements of Abelian Categories[J]. Frontiers of Mathematics in China, 2017, 13(4): 875-892.

[6] Huerta M, Mendoza O, Pérez A. n-Cotorsion Pairs[J]. Journal of Pure and Applied Algebra, 2021, 225(5): 106556-1-106556-34.

[7] Cao W Q, Wei J Q, Wu K L. Recollements and n-Cotorsion Pairs[EB/OL]. (2024-03-07)[2025-08-21]. https://doi.org/10.48550/arXiv:2403.04220.

[8] Franjou V, Pirashvili T. Comparison of Abelian Categories Recollements[J]. Documenta Mathematica, 2004, 9: 41-56.

[9] Psaroudakis C, Vitória J. Recollements of Module Categories[J]. Applied Categorical Structures, 2014, 22(4): 579-593.

[10] Psaroudakis C, Skartsæterhagen Ø, Solberg Ø. Gorenstein Categories, Singular Equivalences and Finite Generation of Cohomology Rings in Recollements[J]. Transactions of the American Mathematical Society (Series B), 2014, 1: 45-95.

[11] Psaroudakis C. Homological Theory of Recollements of Abelian Categories[J]. Journal of Algebra, 2014, 398: 63-110.

[12] Ma X, Xie Z Z, Zhao T W. Support τ-Tilting Modules and Recollement[J]. Colloquium Mathematicum, 2022, 167(2): 303-328.

基金资助

国家自然科学基金(12301042)

安徽工程大学科研启动基金(2022YQQ097)