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

Abstract

Let $$\mathscr {C}$$ be a triangulated category with shift functor [1] and $$\mathcal {R}$$ a contravariantly rigid subcategory of $$\mathscr {C}$$ . We show that a tilting subcategory of $$\mathsf {mod}\,\mathcal {R}$$ lifts to a two-term maximal $$\mathcal {R}[1]$$ -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.