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.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos 11901190 and 11671221), the Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ3205) and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 19B239). The author would like to thank the referee for reading the paper carefully and for many suggestions.
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Communicating Editor: Amalendu Krishna
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Zhou, P. Lifting to two-term relative maximal rigid subcategories in triangulated categories. Proc Math Sci 130, 53 (2020). https://doi.org/10.1007/s12044-020-00568-6
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DOI: https://doi.org/10.1007/s12044-020-00568-6