Auslander–Reiten theory in extriangulated categories
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- by Osamu Iyama, Hiroyuki Nakaoka and Yann Palu
- Trans. Amer. Math. Soc. Ser. B 11 (2024), 248-305
- DOI: https://doi.org/10.1090/btran/159
- Published electronically: January 30, 2024
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Abstract:
The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category $\underline {\mathscr {C}}$ of an extriangulated category $\mathscr {C}$ is a $\tau$-category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if $\mathscr {C}$ has enough projectives, almost split extensions and source morphisms. This gives various consequences on $\underline {\mathscr {C}}$, including Igusa–Todorov’s Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of $\underline {\mathscr {C}}$ via the complete mesh category of the Auslander–Reiten species of $\underline {\mathscr {C}}$. Finally we prove that any locally finite symmetrizable $\tau$-quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms.References
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Bibliographic Information
- Osamu Iyama
- Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
- Address at time of publication: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan
- MR Author ID: 634748
- Email: iyama@ms.u-tokyo.ac.jp
- Hiroyuki Nakaoka
- Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
- MR Author ID: 871532
- Email: nakaoka.hiroyuki@math.nagoya-u.ac.jp
- Yann Palu
- Affiliation: LAMFA, Université de Picardie Jules Verne, 33 rue Saint-Leu, Amiens, France
- MR Author ID: 857813
- Email: yann.palu@u-picardie.fr
- Received by editor(s): September 12, 2021
- Received by editor(s) in revised form: January 27, 2023, and April 17, 2023
- Published electronically: January 30, 2024
- Additional Notes: The first author was supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03923 (B) 22H01113 and (S) 15H05738. The second author was supported by JSPS KAKENHI Grant Numbers JP17K18727 and JP20K03532. The third author was supported by the French ANR grant SC$^3$A (15 CE40 0004 01).
- © Copyright 2024 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 11 (2024), 248-305
- MSC (2020): Primary 16G70, 18E10; Secondary 16E30, 16G50
- DOI: https://doi.org/10.1090/btran/159