论文标题
分层系统和Jordan-Hölder的外侧类别
Stratifying systems and Jordan-Hölder extriangulated categories
论文作者
论文摘要
先前已定义为模块，三角形和确切类别的分层系统，以产生标准分层代数的示例。分层系统$φ$是满足某些正交条件的有限对象。一个非常有趣的属性是，子类别的$ \ MATHCAL {f}（φ）$的对象，允许$φ$中的因子类似于构图系列的过滤，在这些过滤中具有Jordan-H {Ö} lder属性。 本文有两个主要目标。首先，我们介绍了外侧类别的子对象，简单对象和构图系列的概念，以定义Jordan-H {Ö} lder furediangulated类别。此外，我们表征了Jordan-h {Ö} lder，长度，弱小的完整外侧类别，就相关的Grothendieck Monoid和Grothendieck组而言。其次，我们开发了一种在外部类别中分层系统的理论。我们定义了投射分层系统，并表明每个分层系统$φ$在外侧类别中都是最小投影剂的一部分。我们证明$ \ nathcal {f}（φ）$是一个长度，jordan-h {Ö} lder fundiangulated类别当$（φ，q）$满足左侧精确性条件时。 我们给出了几个例子，并回答了否定的Inomoto-Saito的最新问题。
Stratifying systems, which have been defined for module, triangulated and exact categories previously, were developed to produce examples of standardly stratified algebras. A stratifying system $Φ$ is a finite set of objects satisfying some orthogonality conditions. One very interesting property is that the subcategory $\mathcal{F}(Φ)$ of objects admitting a composition series-like filtration with factors in $Φ$ has the Jordan-H{ö}lder property on these filtrations. This article has two main aims. First, we introduce notions of subobjects, simple objects and composition series for an extriangulated category, in order to define a Jordan-H{ö}lder extriangulated category. Moreover, we characterise Jordan-H{ö}lder, length, weakly idempotent complete extriangulated categories in terms of the associated Grothendieck monoid and Grothendieck group. Second, we develop a theory of stratifying systems in extriangulated categories. We define projective stratifying systems and show that every stratifying system $Φ$ in an extriangulated category is part of a minimal projective one $(Φ,Q)$. We prove that $\mathcal{F}(Φ)$ is a length, Jordan-H{ö}lder extriangulated category when $(Φ,Q)$ satisfies a left exactness condition. We give several examples and answer a recent question of Enomoto--Saito in the negative.
