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<p>In this paper, we study triangular matrix categories by using the theory of recollements of abelian categories. Given a triangular matrix category, we construct two canonical recollements. We show that if certain functors of these recollements are exact, then the category appearing in the middle term is actually a category of modules over a triangular matrix category. This result is a generalization of one given by Li (Commun Algebra 46(2):615–628, 2017. <a href="https://doi.org/10.1080/00927872.2017.1327051">https://doi.org/10.1080/00927872.2017.1327051</a>). Finally, we show that if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{Mod}(\mathcal {C})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Mod</mtext> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> admits a nontrivial torsion pair by abelian categories, then <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> is equivalent to a triangular matrix category.</p>

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Characterization of triangular matrix categories via recollements

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月20日

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  • Martha Lizbeth Shaid Sandoval-Miranda,
  • Valente Santiago-Vargas,
  • Edgar Omar Velasco-Páez

摘要

In this paper, we study triangular matrix categories by using the theory of recollements of abelian categories. Given a triangular matrix category, we construct two canonical recollements. We show that if certain functors of these recollements are exact, then the category appearing in the middle term is actually a category of modules over a triangular matrix category. This result is a generalization of one given by Li (Commun Algebra 46(2):615–628, 2017. https://doi.org/10.1080/00927872.2017.1327051). Finally, we show that if \(\textrm{Mod}(\mathcal {C})\) Mod ( C ) admits a nontrivial torsion pair by abelian categories, then \(\mathcal {C}\) C is equivalent to a triangular matrix category.