From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories
From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories
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DOI：
10.1007/s10468-023-10214-0
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发表时间：
2022-01
影响因子：
0.6
通讯作者：
H. Enomoto
中科院分区：
文献类型：
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作者：
H. Enomoto
本文利用完全半分配格上的Kappa映射，用纯格论的方法计算了Abel长范畴中挠类格上的宽子范畴和ICE-闭子范畴的偏序集.对于宽子范畴的偏序集，我们利用正则并表示给出了宽子范畴与扭类之间的双射的两个更简单的构造。更确切地说，对于完全半分配格，我们给出了两个偏序集结构的一组元素的典型连接表示：Kappa顺序（使用扩展的Kappa映射的Barnard-Todorov-Zhu），和核心标签顺序（推广的分片交顺序的一致格）。然后证明了挠类格的偏序集与宽子范畴的偏序集是同构的。作为一个副产品，我们给出了一个简单的描述的碎片交叉顺序有限Coxeter群使用扩展的Kappa映射。
In this paper, we compute the posets of wide subcategories and ICE-closed subcategories from the lattice of torsion classes in an abelian length category in a purely lattice-theoretical way, by using the kappa map in a completely semidistributive lattice. As for the poset of wide subcategories, we give two more simple constructions via a bijection between wide subcategories and torsion classes with canonical join representations. More precisely, for a completely semidistributive lattice, we give two poset structures on the set of elements with canonical join representations: the kappa order (defined using the extended kappa map of Barnard–Todorov–Zhu), and the core label order (generalizing the shard intersection order for congruence-uniform lattices). Then we show that these posets for the lattice of torsion classes coincide and are isomorphic to the poset of wide subcategories. As a byproduct, we give a simple description of the shard intersection order on a finite Coxeter group using the extended kappa map.