Relative Auslander bijection in \(n\)-exangulated categories. (English) Zbl 07729522
Summary: The aim of this article is to study the relative Auslander bijection in \(n\)-exangulated categories. More precisely, we introduce the notion of generalized Auslander-Reiten-Serre duality and exploit a bijection triangle, which involves the generalized Auslander-Reiten-Serre duality and the restricted Auslander bijection relative to the subfunctor. As an application, this result generalizes the work by Zhao in extriangulated categories.
MSC:
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 18G80 | Derived categories, triangulated categories |
| 18E10 | Abelian categories, Grothendieck categories |
Keywords:
\(n\)-exangulated category; generalized Auslander-Reiten-Serre duality; restricted Auslander bijectionReferences:
| [1] | Auslander, M., Functors and morphisms determined by objects. Representation Theory of Algebras, 1-244 (1978), New York: Marcel Dekker, New York · Zbl 0383.16015 |
| [2] | Auslander, M.; Reiten, I., Representation theory of Artin algebras. III: Almost split sequences, Commun. Algebra, 3, 239-294 (1975) · Zbl 0331.16027 · doi:10.1080/00927877508822046 |
| [3] | Auslander, M.; Reiten, I., Representation theory of Artin algebras. IV: Invariants given by almost split sequences, Commun. Algebra, 5, 443-518 (1977) · Zbl 0396.16007 · doi:10.1080/00927877708822180 |
| [4] | Chen, X-W, The Auslander bijections and universal extensions, Ark. Mat., 55, 41-59 (2017) · Zbl 1388.16018 · doi:10.4310/ARKIV.2017.v55.n1.a2 |
| [5] | Dräxler, P.; Reiten, I.; Smalø, S. O.; Solberg, Exact categories and vector space categories, Trans. Am. Math. Soc., 351, 647-682 (1999) · Zbl 0916.16002 · doi:10.1090/S0002-9947-99-02322-3 |
| [6] | Geiss, C.; Keller, B.; Oppermann, S., n-angulated categories, J. Reine Angew. Math., 675, 101-120 (2013) · Zbl 1271.18013 |
| [7] | J. He, J. He, P. Zhou: Auslander-Reiten-Serre duality for n-exangulated categories. Available at https://arxiv.org/abs/2112.00981 (2021), 24 pages. |
| [8] | J. He, J. He, P. Zhou: Higher Auslander-Reiten sequences via morphisms determined by objects. Available at https://arxiv.org/abs/2111.06522 (2021), 28 pages. |
| [9] | He, J.; Hu, J.; Zhang, D.; Zhou, P., On the existence of Auslander-Reiten n-exangles in n-exangulated categories, Ark. Mat., 60, 365-385 (2022) · Zbl 1509.18006 · doi:10.4310/ARKIV.2022.v60.n2.a8 |
| [10] | J. He, P. Zhou: n-exact categories arising from n-exangulated categories. Available at https://arxiv.org/abs/2109.12954 (2021), 16 pages. |
| [11] | Herschend, M.; Liu, Y.; Nakaoka, H., n-exangulated categories. I: Definitions and fundamental properties, J. Algebra, 570, 531-586 (2021) · Zbl 1506.18015 · doi:10.1016/j.jalgebra.2020.11.017 |
| [12] | Herschend, M.; Liu, Y.; Nakaoka, H., n-exangulated categories. II: Constructions from n-cluster tilting subcategories, J. Algebra, 594, 636-684 (2022) · Zbl 1531.18013 · doi:10.1016/j.jalgebra.2021.11.042 |
| [13] | Hu, J.; Zhang, D.; Zhou, P., Two new classes of n-exangulated categories, J. Algebra, 568, 1-21 (2021) · Zbl 1458.18006 · doi:10.1016/j.jalgebra.2020.09.041 |
| [14] | O. Iyama, H. Nakaoka, Y. Palu: Auslander-Reiten theory in extriangulated categories. Available at https://arxiv.org/abs/1805.03776v2 (2019), 40 pages. |
| [15] | Jasso, G., n-abelian and n-exact categories, Math. Z., 283, 703-759 (2016) · Zbl 1356.18005 · doi:10.1007/s00209-016-1619-8 |
| [16] | Jiao, P., The generalized Auslander-Reiten duality on an exact category, J. Algebra Appl., 17, 14 (2018) · Zbl 1407.16014 · doi:10.1142/S0219498818502274 |
| [17] | Jiao, P., Auslander’s defect formula and a commutative triangle in an exact category, Front. Math. China, 15, 115-125 (2020) · Zbl 1457.16014 · doi:10.1007/s11464-020-0814-4 |
| [18] | Liu, Y.; Zhou, P., Frobenius n-exangulated categories, J. Algebra, 559, 161-183 (2020) · Zbl 1448.18022 · doi:10.1016/j.jalgebra.2020.03.036 |
| [19] | Nakaoka, H.; Palu, Y., Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég., 60, 117-193 (2019) · Zbl 1451.18021 |
| [20] | Ringel, C. M., The Auslander bijections: How morphisms are determined by modules, Bull. Math. Sci., 3, 409-484 (2013) · Zbl 1405.16030 · doi:10.1007/s13373-013-0042-2 |
| [21] | T. Zhao: Relative Auslander bijection in extriangulated categories. Available at www.researchgate.net/publication/367217942 (2021). |
| [22] | Zhao, T.; Tan, L.; Huang, Z., Almost split triangles and morphisms determined by objects in extriangulated categories, J. Algebra, 559, 346-378 (2020) · Zbl 1440.18024 · doi:10.1016/j.jalgebra.2020.04.021 |
| [23] | Zhao, T.; Tan, L.; Huang, Z., A bijection triangle in extriangulated categories, J. Algebra, 574, 117-153 (2021) · Zbl 1459.18007 · doi:10.1016/j.jalgebra.2020.12.043 |
| [24] | Zheng, Q.; Wei, J., (n + 2)-angulated quotient categories, Algebra Colloq., 26, 689-720 (2019) · Zbl 1427.18006 · doi:10.1142/S1005386719000506 |
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