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Largest exact structures and almost split sequences on hearts of twin cotorsion pairs

Part of: Abelian categories

Published online by Cambridge University Press:  15 April 2024

Yu Liu ,
Wuzhong Yang  and
Panyue Zhou
Yu Liu
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, 710062 Xi’an, Shaanxi, P. R. China e-mail: [email protected]
Wuzhong Yang*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, 401331 Chongqing, P. R. China
Panyue Zhou
Affiliation:
School of Mathematics and Statistics, Changsha University of Science and Technology, 410114 Changsha, Hunan, P. R. China e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Hearts of cotorsion pairs on extriangulated categories are abelian categories. On the other hand, hearts of twin cotorsion pairs are not always abelian. They were shown to be semi-abelian by Liu and Nakaoka. Moreover, Hassoun and Shah proved that they are quasi-abelian under certain conditions. In this article, we first show that the heart of any twin cotorsion pair has a largest exact category structure and is always quasi-abelian. We also provide a sufficient and necessary condition for the heart of a twin cotorsion pair being abelian. Then by using the results we have got, we investigate the almost split sequences in the hearts of twin cotorsion pairs. Finally, as an application, we show that a Krull–Schmidt, Hom-finite triangulated category has a Serre functor whenever it has a cluster tilting object.

Keywords

Twin cotorsion pairheartquasi-abelianalmost split sequenceSerre functor

MSC classification

Primary: 18E10: Exact categories, abelian categories
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Yu Liu is supported by the National Natural Science Foundation of China (Grant No. 12171397). Wuzhong Yang is supported by the National Natural Science Foundation of China (Grant No. 12271321) and the Scientific Research Foundation of Chongqing Normal University (Grant No. 24XLB008). Panyue Zhou is supported by the National Natural Science Foundation of China (Grant No. 12371034) and by the Hunan Provincial Natural Science Foundation of China (Grant No. 2023JJ30008). Wuzhong Yang is the corresponding author.

References

Auslander, M. and Reiten, I., Representation theory of Artin algebras. III. Almost split sequences . Comm. Algebra 3(1975), 239–294.10.1080/00927877508822046CrossRefGoogle Scholar
Auslander, M. and Reiten, I., Representation theory of Artin algebras. IV. Invariants given by almost split sequences . Comm. Algebra 5(1977), no. 5, 443–518.10.1080/00927877708822180CrossRefGoogle Scholar
Beilinson, A., Bernstein, J., and Deligne, P., Faisceaux pervers, (French) [Perverse sheaves] Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, 100, Société mathématique de France, Paris, 1982, pp. 5–171.Google Scholar
Bonet, J. and Dierolf, S., The pullback for bornological and ultrabornological spaces . Note Mat. 25(2005/06), no. 1, 63–67.Google Scholar
Buan, A. B. and Marsh, R., From triangulated categories to module categories via localization II: Calculus of fractions . J. Lond. Math. Soc. (2) 86(2012), no. 1, 152–170.10.1112/jlms/jdr077CrossRefGoogle Scholar
Buan, A. B., Marsh, R., Reineke, M., Reiten, I., and Todorov, G., Tilting theory and cluster combinatorics . Adv. Math. 204(2006), no. 2, 572–618.10.1016/j.aim.2005.06.003CrossRefGoogle Scholar
Buan, A. B., Marsh, R., and Reiten, I., Cluster-tilted algebras . Trans. Amer. Math. Soc. 359(2007), no 1, 323–332.10.1090/S0002-9947-06-03879-7CrossRefGoogle Scholar
Bühler, T., Exact categories . Expo. Math. 28(2010), 1–69.10.1016/j.exmath.2009.04.004CrossRefGoogle Scholar
Crivei, S., Maximal exact structures on additive categories revisited . Math. Nachr. 285(2012), no. 4, 440–446.10.1002/mana.201000065CrossRefGoogle Scholar
Demonet, L. and Liu, Y., Quotients of exact categories by cluster tilting subcategories as module categories . J. Pure Appl. Algebra 217(2013), no 12, 2282–2297.10.1016/j.jpaa.2013.03.007CrossRefGoogle Scholar
Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988, x+208 pp.10.1017/CBO9780511629228CrossRefGoogle Scholar
Hassoun, S. and Shah, A., Integral and quasi-abelian hearts of twin cotorsion pairs on extriangulated categories . Comm. Algebra 48(2020), no. 12, 5142–5162.10.1080/00927872.2020.1779737CrossRefGoogle Scholar
Herschend, M., Liu, Y., and Nakaoka, H., $n$ -exangulated categories (I): Definitions and fundamental properties . J. Algebra 570(2021), 531–586.10.1016/j.jalgebra.2020.11.017CrossRefGoogle Scholar
Huang, Q. and Zhou, P., Abelian hearts of twin cotorsion pairs on extriangulated categories . Algebra Colloq. 30(2023), no. 3, 449–466.10.1142/S1005386723000378CrossRefGoogle Scholar
Iyama, O., Nakaoka, H., and Palu, Y., Auslander–Reiten theory in extriangulated categories . Trans. Amer. Math. Soc. Ser. B 11(2024), 248–305.10.1090/btran/159CrossRefGoogle Scholar
Iyama, O. and Yoshino, Y., Mutation in triangulated categories and rigid Cohen–Macaulay modules . Invent. Math. 172(2008), no. 1, 117–168.10.1007/s00222-007-0096-4CrossRefGoogle Scholar
Janelidze, G., Márki, L., and Tholen, W., Semi-abelian categories . J. Pure Appl. Algebra 168(2002), nos. 2–3, 367–386.10.1016/S0022-4049(01)00103-7CrossRefGoogle Scholar
Keller, B. and Reiten, I., Cluster-tilted algebras are Gorenstein and stably Calabi–Yau . Adv. Math. 211(2007), 123–151.10.1016/j.aim.2006.07.013CrossRefGoogle Scholar
Koenig, S. and Zhu, B., From triangulated categories to abelian categories: cluster tilting in a general framework . Math. Z. 258(2008), no. 1, 143–160.10.1007/s00209-007-0165-9CrossRefGoogle Scholar
Liu, S., Auslander–Reiten theory in a Krull–Schmidt category . São Paulo J. Math. Sci. 4(2010), no. 3, 425–472.10.11606/issn.2316-9028.v4i3p425-472CrossRefGoogle Scholar
Liu, Y., Hearts of twin cotorsion pairs on exact categories . J. Algebra 394(2013), 245–284.10.1016/j.jalgebra.2013.07.028CrossRefGoogle Scholar
Liu, Y., Abelian hearts of twin cotorsion pairs . Arch. Math. 115(2020), 379–389.10.1007/s00013-020-01481-0CrossRefGoogle Scholar
Liu, Y. and Nakaoka, H., Hearts of twin cotorsion pairs on extriangulated categories . J. Algebra 528(2019), 96–149.10.1016/j.jalgebra.2019.03.005CrossRefGoogle Scholar
Nakaoka, H., General heart construction on a triangulated category (I): Unifying $t$ -structures and cluster tilting subcategories . Appl. Categ. Structures 19(2011), no. 6, 87–899.10.1007/s10485-010-9223-2CrossRefGoogle Scholar
Nakaoka, H., General heart construction for twin torsion pairs on triangulated categories . J. Algebra 374(2013), 195–215.10.1016/j.jalgebra.2012.10.027CrossRefGoogle Scholar
Nakaoka, H. and Palu, Y., Extriangulated categories, Hovey twin cotorsion pairs and model structures . Cah. Topol. Géom. Différ. Catég. 60(2019), no. 2, 117–193.Google Scholar
Quillen, D., Higher algebraic K-theory: I . In: Higher K-theories, Proceedings Conference, Battelle Memorial Institute, Seattle, Washington, 1972, Lecture Notes in Mathematics, 341, Springer, Berlin, 1973, pp. 85–147.10.1007/BFb0067053CrossRefGoogle Scholar
Reiten, I. and Van Den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality . J. Amer. Math. Soc. 15(2002), no. 2, 295–366.10.1090/S0894-0347-02-00387-9CrossRefGoogle Scholar
Rump, W., Almost abelian categories . Cah. Topol. Géom. Différ. Catég. 42(2001), no. 3, 163–225.Google Scholar
Rump, W., A counterexample to Raikov’s conjecture . Bull. Lond. Math. Soc. 40(2008), no. 6, 985–994.10.1112/blms/bdn080CrossRefGoogle Scholar
Salce, L., Cotorsion theories for abelian groups . In: Symposia Mathematica, vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), Academic Press, London–New York, 1979, pp. 11–32.Google Scholar
Shah, A., Auslander–Reiten theory in quasi-abelian and Krull–Schmidt categories . J. Pure Appl. Algebra 224(2020), no. 1, 98–124.10.1016/j.jpaa.2019.04.017CrossRefGoogle Scholar
Sieg, D. and Wegner, S. A., Maixmal exact structures on additive categories . Math. Nachr. 284(2011), no. 16, 2093–2100.10.1002/mana.200910154CrossRefGoogle Scholar
Šťovíček, J., Exact model categories, approximation theory, and cohomology of quasi-coherent sheaves . In: Advances in representation theory of algebras, EMS Ser. Congr. Rep., European Mathematical Society, Zürich, 2013, pp. 297–367.Google Scholar
Zhou, P. and Zhu, B., Triangulated quotient categories revisited . J. Algebra 502(2018), 196–232.10.1016/j.jalgebra.2018.01.031CrossRefGoogle Scholar
Zhou, P. and Zhu, B., Cluster-tilting subcategories in extriangulated categories . Theory Appl. Categ. 34(2019), Article no. 8, 221–242.Google Scholar