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Triangular matrix categories over quasi-hereditary categories

Part of: Homological algebra Abelian categories

Published online by Cambridge University Press:  21 March 2024

Rafael Francisco Ochoa De La Cruz ,
Martin Ortíz Morales  and
Valente Santiago Vargas
Rafael Francisco Ochoa De La Cruz
Affiliation:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Mexico City, CP 04510, Mexico
Martin Ortíz Morales
Affiliation:
Facultad de Ciencias, Universidad Autónoma del Estado de México, Campus Universitario “El Cerrillo, Piedras Blancas”, Carretera Toluca-Ixtlahuaca Km. 15.5, Toluca, CP 50200, Mexico
Valente Santiago Vargas*
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Mexico City, CP 04510, Mexico
*
Corresponding author: Valente Santiago Vargas; valente.santiago.v@gmail.com
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Abstract

In this paper, we prove that the lower triangular matrix category $\Lambda =\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right ]$, where $\mathcal{T}$ and $\mathcal{U}$ are $\textrm{Hom}$-finite, Krull–Schmidt $K$-quasi-hereditary categories and $M$ is an $\mathcal{U}\otimes _K \mathcal{T}^{op}$-module that satisfies suitable conditions, is quasi-hereditary. This result generalizes the work of B. Zhu in his study on triangular matrix algebras over quasi-hereditary algebras. Moreover, we obtain a characterization of the category of the $_\Lambda \Delta$-filtered $\Lambda$-modules.

Keywords

Functor CategoriesMatrix CategoriesPath CategoriesQuasihereditary Categories

MSC classification

Primary: 18E10: Exact categories, abelian categories
Secondary: 18G25: Relative homological algebra, projective classes

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 3 , September 2024 , pp. 449 - 470
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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