Wide Subcategories and Lattices of Torsion Classes

By Sotaro Asai, C. Pfeifer. Algebras and Representation Theory, 2019

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In this paper, we study the relationship between wide subcategories and torsion classes of an abelian length category A $\mathcal {A}$ from the point of view of lattice theory. Motivated by τ -tilting reduction of Jasso, we mainly focus on intervals [ U , T ] $[\mathcal {U},\mathcal {T}]$ in the lattice tors A $\operatorname {\mathsf {tors}} \mathcal {A}$ of torsion classes in A $\mathcal {A}$ such that W : = U ⊥ ∩ T $\mathcal {W}:=\mathcal {U}^{\perp } \cap \mathcal {T}$ is a wide subcategory of A $\mathcal {A}$ ; we call these intervals wide intervals. We prove that a wide interval [ U , T ] $[\mathcal {U},\mathcal {T}]$ is isomorphic to the lattice tors W $\operatorname {\mathsf {tors}} \mathcal {W}$ of torsion classes in the abelian category W $\mathcal {W}$ . 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12e1231af99d6813876630d5291a88333e88e628

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In Algebra and Number Theory

  1. Sotaro Asai
  2. C. Pfeifer

In Geometry and Topology

  1. Sotaro Asai
  2. C. Pfeifer

In Mathematical Physics

  1. Sotaro Asai
  2. C. Pfeifer

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