Classification of split torsion torsionfree triples in module categories

无点版本的无挠类与 Goldie 挠理论

A point-free version of torsionfree classes and the Goldie torsion theory

本文利用挠理论的对称性，在 idiom 上定义 division free 集与非奇异区间，并建立其与奇异区间除法集的典范联系，为无点非奇异理论奠定部分基础。

Torsionfree Dimension of Modules and Self-Injective Dimension of Rings

Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n\geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_R^i(M, R)=0$ for any $1…

平衡对、几乎格罗蒙德环和余 torsion 三元组

Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples

本文研究了环R中平衡对与余 torsion 三元组的关系，证明了右几乎格罗蒙德环的条件，并通过无限维翻转理论分类了所有余 torsion 三元组。

关于有限生成投射模复形的 Auslander-Bridger-Yoshino 理论

On the Auslander-Bridger-Yoshino theory for complexes of finitely generated projective modules

本文基于 Auslander-Bridger 稳定模理论与 Yoshino 的稳定复形理论，引入了 *torsionfreeness 和 *reflexivity 等概念的高阶版本，并证明了有限生成投射模复形的 Auslander-Bridger 逼近定理。

Torsion pairs in categories of modules over a preadditive category

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categori…

On $τ$-tilting subcategories

The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of $τ$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $τ$-tilting subcategories of $\mathscr{A}$, generalizing the bijection by Bauer, Botnan, Oppermann and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr{A}$. General definitions and results are exemplified using persistent mod…