Articles written in Proceedings – Mathematical Sciences

    • Abelian quotients of extriangulated categories


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      We prove that certain subquotient categories of extriangulated categories are abelian. As a particular case, if an extriangulated category $\mathcal{C}$ has a cluster-tilting subcategory $\mathcal{X}$, then $\mathcal{C}/\mathcal{X}$ is abelian. This unifies a result by Koenig and Zhu (Math. Z. 258 (2008) 143-160) for triangulated categories and a result by Demonet and Liu (J. Pure Appl. Algebra 217(12) (2013) 2282-2297) for exact categories.

    • Lifting to two-term relative maximal rigid subcategories in triangulated categories


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      Let $\mathscr{C}$ be a triangulated category with shift functor [1] and $\mathscr{R}$ a contravariantly rigid subcategory of $\mathscr{C}$ . We show that a tilting subcategory of $\mod\mathscr{ R}$ lifts to a two-term maximal $\mathscr{R}$[1]-rigid subcategory of $\mathscr{C}$ . As an application, our result generalizes a result by Xie and Liu (Proc. Amer. Math. Soc. 141(10) (2013) 3361–3367) for maximal rigid objects and a result by Fu and Liu (Comm. Algebra 37(7) (2009) 2410–2418) for cluster tilting objects.

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