关于余挠三元组的余星子范畴和余倾斜子范畴

设$mathcal{A}$ 是一个Abel范畴,且 $(mathcal{X}, mathcal{Z},mathcal{Y})$ 是一个完全遗传余挠三元组.介绍 $mathcal{A}$ 的 $n$-$mathcal{Y}$-余倾斜子范畴的定义,并给出 $n$-$mathcal{Y}$-余倾斜子范畴的一个刻画,类似于 $n$-余倾斜模的 Bazzoni 刻画.作为应用,证明了在一个几乎 Gorenstein 环 $R$ 上, 如果 $mathcal{GP}$ 是 $n$-$mathcal{GI}$-余倾斜的, 那么 $R$ 是一个 $n$-Gorenstein 环, 其中 $mathcal{GP}$ 表示 Gorenstein 投射 $R$-模组成的子范畴且 $mathcal{GI}$ 表示 Gorenstein 内射 $R$-模组成的子范畴. 进而, 研究 任意环$R$上的$n$-余星子范畴, 以及关于余挠三元组 $(mathcal{P}, R$-Mod, $mathcal{I})$ 的 $n$-$mathcal{I}$-子范畴与 $n$-余星子范畴之间的关系, 其中 $mathcal{P}$ 表示投射左 $R$-模组成的子范畴且 $mathcal{I}$ 表示内射左 $R$-模组成的子范畴.

Costar Subcategories and Cotilting Subcategories with Respect to Cotorsion Triples

Let $mathcal{A}$ be an abelian category, and $(mathcal{X}, mathcal{Z},mathcal{Y})$ be a complete hereditary cotorsion triple. We introduce the definition of $n$-$mathcal{Y}$-cotilting subcategories of $mathcal{A}$, and give a characterization of $n$-$mathcal{Y}$-cotilting subcategories, which is similar to Bazzoni characterization of $n$-cotilting modules. As an application, we prove that if $mathcal{GP}$ is $n$-$mathcal{GI}$-cotilting over a virtually Gorenstein ring $R$, then $R$ is an $n$-Gorenstein ring, where $mathcal{GP}$ denotes the subcategory of Gorenstein projective $R$-modules and $mathcal{GI}$ denotes the subcategory of Gorenstein injective $R$-modules. Furthermore, we investigate $n$-costar subcategories over arbitrary ring $R$, and the relationship between $n$-$mathcal{I}$-cotilting subcategories with respect to cotorsion triple $(mathcal{P}, R$-Mod, $mathcal{I})$ and $n$-costar subcategories, where $mathcal{P}$ denotes the subcategory of projective left $R$-modules and $mathcal{I}$ denotes the subcategory of injective left $R$-modules.