论文标题
来自Grothendieck类别的有限类型对象的扭转和无扭转类
Torsion and torsion-free classes from objects of finite type in Grothendieck categories
论文作者
论文摘要
在任意的Grothendieck类别中，我们发现了$ \ text {fp} _n $ imptive对象作为扭转类的必要条件。通过这样做，我们提出了$ n $ herseditary类别的概念。我们还用Grothendieck类别中的$ \ text {fp} _n $ -flat对象定义和研究了具有一组小型投影对象集的类别，并为此类别提供了多种等效条件，使该类无扭转。最后，我们在上下文模块中介绍了几个应用程序和示例，这些应用程序和$ n $ herseditart类别的链，来自添加剂类别的模块的链条和类别的链条复合物以及阿贝尔群体类别。关于后一种设置，我们发现这些函数类别何时在域添加剂类别方面是$ n $ hersedarity的表征。
In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-injective objects to be a torsion class. By doing so, we propose a notion of $n$-hereditary categories. We also define and study the class of $\text{FP}_n$-flat objects in Grothendieck categories with a generating set of small projective objects, and provide several equivalent conditions for this class to be torsion-free. In the end, we present several applications and examples of $n$-hereditary categories in the contexts modules over a ring, chain complexes of modules and categories of additive functors from an additive category to the category of abelian groups. Concerning the latter setting, we find a characterization of when these functor categories are $n$-hereditary in terms of the domain additive category.
