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From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories

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Abstract

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.

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Acknowledgements

The author would like to thank Osamu Iyama for sharing him the proof of Lemma 4.26. He would also like to thank Yuya Mizuno for helpful discussions. Additionally, the author would like to express their gratitude to the anonymous referee for their careful reading and valuable comments. This work is supported by JSPS KAKENHI Grant Number JP21J00299.

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  1. Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka, 599-8531, Japan

    Haruhisa Enomoto

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Correspondence to Haruhisa Enomoto.

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Enomoto, H. From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories. Algebr Represent Theor 26, 3223–3253 (2023). https://doi.org/10.1007/s10468-023-10214-0

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