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Wide Subcategories and Lattices of Torsion Classes

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In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category \(\mathcal {A}\) from the point of view of lattice theory. Motivated by τ-tilting reduction of Jasso, we mainly focus on intervals \([\mathcal {U},\mathcal {T}]\) in the lattice \(\operatorname {\mathsf {tors}} \mathcal {A}\) of torsion classes in \(\mathcal {A}\) such that \(\mathcal {W}:=\mathcal {U}^{\perp } \cap \mathcal {T}\) is a wide subcategory of \(\mathcal {A}\); we call these intervals wide intervals. We prove that a wide interval \([\mathcal {U},\mathcal {T}]\) is isomorphic to the lattice \(\operatorname {\mathsf {tors}} \mathcal {W}\) of torsion classes in the abelian category \(\mathcal {W}\). We also characterize wide intervals in two ways: First, in purely lattice theoretic terms based on the brick labeling established by Demonet–Iyama–Reading–Reiten–Thomas; and second, in terms of the Ingalls–Thomas correspondences between torsion classes and wide subcategories, which were further developed by Marks–Šťovíček.

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The authors thank Aaron Chan, Laurent Demonet, Haruhisa Enomoto, Osamu Iyama, Gustavo Jasso and Jan Schröer for kind instructions and discussions. The second named author would like to thank the Mathematical Institute of the University of Bonn, where most of his work was done as part of his Master studies.


The first named author was supported by Japan Society for the Promotion of Science KAKENHI JP16J02249, JP19K14500 and JP20J00088.

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Correspondence to Sota Asai.

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Presented by: Henning Krause

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Asai, S., Pfeifer, C. Wide Subcategories and Lattices of Torsion Classes. Algebr Represent Theor 25, 1611–1629 (2022).

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