On the stable equivalences between finite tensor categories

Xu, Yuying; Liu, Gongxiang

doi:10.1090/proc/16283

On the stable equivalences between finite tensor categories
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by Yuying Xu and Gongxiang Liu;

Proc. Amer. Math. Soc. 151 (2023), 1867-1876

DOI: https://doi.org/10.1090/proc/16283

Published electronically: February 10, 2023

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Abstract:

We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk \text {-}$linear monoidal functor can be lifted to a tensor equivalence under some certain conditions.

References

Nicolás Andruskiewitsch, Pavel Etingof, and Shlomo Gelaki, Triangular Hopf algebras with the Chevalley property, Michigan Math. J. 49 (2001), no. 2, 277–298. MR 1852304, DOI 10.1307/mmj/1008719774
Maurice Auslander, Comments on the functor $\textrm {Ext}$, Topology 8 (1969), 151–166. MR 237606, DOI 10.1016/0040-9383(69)90006-8
Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
Paul Balmer, Tensor triangular geometry, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 85–112. MR 2827786
Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168. MR 2196732, DOI 10.1515/crll.2005.2005.588.149
Michel Broué, Equivalences of blocks of group algebras, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 1–26. MR 1308978, DOI 10.1007/978-94-017-1556-0_{1}
Pavel Etingof and Viktor Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004), no. 3, 627–654, 782–783 (English, with English and Russian summaries). MR 2119143, DOI 10.17323/1609-4514-2004-4-3-627-654
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, Providence, RI, 2015. MR 3242743, DOI 10.1090/surv/205
Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
Yevgenia Kashina, Susan Montgomery, and Siu-Hung Ng, On the trace of the antipode and higher indicators, Israel J. Math. 188 (2012), 57–89. MR 2897723, DOI 10.1007/s11856-011-0092-7
D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. IV, J. Amer. Math. Soc. 7 (1994), no. 2, 383–453. MR 1239507, DOI 10.1090/S0894-0347-1994-1239507-1
Markus Linckelmann, Stable equivalences of Morita type for self-injective algebras and $p$-groups, Math. Z. 223 (1996), no. 1, 87–100. MR 1408864, DOI 10.1007/PL00004556
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
D. K. Nakano, K. B. Vashaw, and M. T. Yakimov, Noncommutative tensor triangular geometry, preprint arXiv:1909.04304v6, 2021.
D. K. Nakano, K. B. Vashaw, and M. T. Yakimov, On the spectrum and support theory of a finite tensor category, preprint arXiv:2112.11170v3, 2022.
Siu-Hung Ng and Peter Schauenburg, Central invariants and higher indicators for semisimple quasi-Hopf algebras, Trans. Amer. Math. Soc. 360 (2008), no. 4, 1839–1860. MR 2366965, DOI 10.1090/S0002-9947-07-04276-6