Monobrick, a uniform approach to torsion-free classes and wide subcategories
Monobrick, a uniform approach to torsion-free classes and wide subcategories
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Monobrick，无扭转类别和广泛子类别的统一方法
DOI：
10.1016/j.aim.2021.108113
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发表时间：
2021
影响因子：
1.7
通讯作者：
Enomoto Haruhisa
中科院分区：
文献类型：
--
作者：
Enomoto Haruhisa;Sakai Arashi;Enomoto Haruhisa;Enomoto Haruhisa
对于一个长度阿贝尔范畴，我们证明了所有的挠自由类可以只使用砖的信息进行分类，包括非功能有限的。我们的想法是考虑一个无扭类中的简单对象集，它具有以下属性：它是一组砖，其中它们之间的每个非零映射都是一个注入。我们称这样的集合为单块体。在本文中，我们提供了一个统一的方法来研究挠自由类和宽子范畴通过monobrick。我们证明了Monobrick与左Schur子范畴是双射的，它包含了所有在扩张、核和象下闭的子范畴，从而统一了无挠类和宽子范畴。然后，我们证明了无挠类双射对应于共尾闭单块。在不使用τ-倾斜理论的情况下，利用Monobrick给出了长交换范畴中挠（-free）类和宽子范畴的几个已知结果（如有限性结果和双射）.对于Nakayama代数，左Schur子范畴与扩张、核和象闭的子范畴是相同的，并且证明了左Schur子范畴的个数与大Schröder数有关.
For a length abelian category, we show that all torsion-free classes can be classified by using only the information on bricks, including non functorially-finite ones. The idea is to consider the set of simple objects in a torsion-free class, which has the following property: it is a set of bricks where every non-zero map between them is an injection. We call such a set a monobrick. In this paper, we provide a uniform method to study torsion-free classes and wide subcategories via monobricks. We show that monobricks are in bijection with left Schur subcategories, which contains all subcategories closed under extensions, kernels and images, thus unifies torsion-free classes and wide subcategories. Then we show that torsion-free classes bijectively correspond to cofinally closed monobricks. Using monobricks, we deduce several known results on torsion(-free) classes and wide subcategories (e.g. finiteness result and bijections) in length abelian categories, without usingτ-tilting theory. For Nakayama algebras, left Schur subcategories are the same as subcategories closed under extensions, kernels and images, and we show that its number is related to the large Schröder number.