Hostname: page-component-5cf477f64f-zrtmk Total loading time: 0 Render date: 2025-04-01T14:35:12.369Z Has data issue: false hasContentIssue false
  • English
  • Français

AUSLANDER–IYAMA CORRESPONDENCE FOR EXACT DG CATEGORIES

Part of: Abelian categories Homological algebra Homological methods

Published online by Cambridge University Press:  17 February 2025

XIAOFA CHEN Open the ORCID record for XIAOFA CHEN [Opens in a new window]
XIAOFA CHEN*
Affiliation:
School of Mathematical Sciences University of Science and Technology of China Hefei JinZhai Road No. 96 P. R. China cxf2011@mail.ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the Auslander–Iyama correspondence to the setting of exact dg categories. By specializing it to exact dg categories concentrated in degree zero, we obtain a generalization of the higher Auslander correspondence for exact categories due to Ebrahimi–Nasr-Isfahani (in the case of exact categories with split retractions).

Keywords

extriangulated categoryd-Auslander categoryexact dg categoryderived dg categoryAuslander–Iyama correspondence

MSC classification

Primary: 18G35: Chain complexes
Secondary: 18E20: Embedding theorems 16E35: Derived categories
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 25
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aihara, T. and Iyama, O., Silting mutation in triangulated categories , J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668.CrossRefGoogle Scholar
Auslander, M., Reiten, I. and Smalø, S., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, 1995. (English). https://doi.org/10.1017/CBO9780511623608.CrossRefGoogle Scholar
Auslander, M., Representation dimension of Artin algebras, Lecture Notes, Queen Mary College, London, 1971, reprinted in: Selected Works of Maurice Auslander, part 1, edited and with a foreword by Idun Reiten, Sverre O. Smalø, and Øyvind Solberg, Amer. Math. Soc., Providence, 1999.Google Scholar
Auslander, M. and Solberg, Ø., Gorenstein algebras and algebras with dominant dimension at least $2$ , Comm. Algebra 21 (1993), no. 11, 3897–3934.CrossRefGoogle Scholar
Beilinson, A. A., Bernstein, J. and Deligne, P., Analyse et topologie sur les espaces singuliers, Astérisque, 100, Société Mathématique de France, 1982 (French).Google Scholar
Beligiannis, A., Relative homology, higher cluster-tilting theory and categorified Auslander–Iyama correspondence , J. Algebra 444 (2015), 367–503.CrossRefGoogle Scholar
Bühler, T., Exact categories , Expo. Math. 28 (2010), no. 1, 1–69.CrossRefGoogle Scholar
Chen, X., On exact dg categories. Ph.D. thesis, Université Paris Cité, 2023.Google Scholar
Chen, X., Exact dg categories I: Foundations, preprint, arXiv:2402.10694 [math.RT], 2024.Google Scholar
Chen, X., Exact dg categories II: The embedding theorem, preprint, arXiv:2406.11226 [math.RT], 2024.Google Scholar
Chen, X., Iyama–Solberg correspondence for exact dg categories, preprint, arXiv:2401.02064 [math.RT], 2024.Google Scholar
Ebrahimi, R. and Nasr-Isfahani, A., Higher Auslander correspondence for exact categories, preprint, arXiv:2108.13645 [math.RT], 2021.Google Scholar
Fang, X., Gorsky, M., Palu, Y., Plamondon, P.-G. and Pressland, M., Extriangulated ideal quotients, with applications to cluster theory and gentle algebras, preprint, arXiv:2308.05524 [math.RT], 2023.Google Scholar
Grevstad, J. F., Higher Auslander–Solberg correspondence for exact categories, J. Pure Appl. Algebra 228 (2024), no. 6, Paper No. 107603, 23. https://doi.org/10.1016/j.jpaa.2023.107603.CrossRefGoogle Scholar
Gorsky, M., Nakaoka, H. and Palu, Y., Positive and negative extensions in extriangulated categories, preprint, arXiv:2103.12482 [math.CT], 2021.Google Scholar
Gorsky, M., Nakaoka, H. and Palu, Y., Hereditary extriangulated categories: Silting objects, mutation, negative extensions, preprint, arXiv:2303.07134 [math.RT], 2023.Google Scholar
Henrard, R., Kvamme, S. and van Roosmalen, A.-C., Auslander’s formula and correspondence for exact categories , Adv. Math. 401 (2022), Paper No. 108296, 65.CrossRefGoogle Scholar
Iyama, O., Auslander correspondence , Adv. Math. 210 (2007), no. 1, 51–82.CrossRefGoogle Scholar
Iyama, O., Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories , Adv. Math. 210 (2007), no. 1, 22–50.CrossRefGoogle Scholar
Iyama, O. and Solberg, Ø., Auslander–Gorenstein algebras and precluster tilting , Adv. Math. 326 (2018), 200–240.CrossRefGoogle Scholar
Jasso, G., $n$ -Abelian and $n$ -exact categories , Math. Z. 283 (2016), no. 3–4, 703–759.CrossRefGoogle Scholar
Klapproth, C., $n$ -Extension closed subcategories of $n$ -exangulated categories, preprint, arXiv:2209.01128, 2022.Google Scholar
Liu, Y. and Nakaoka, H., Hearts of twin cotorsion pairs on extriangulated categories , J. Algebra 528 (2019), 96–149.CrossRefGoogle Scholar
Msapato, D., The Karoubi envelope and weak idempotent completion of an extriangulated category , Appl. Categ. Struct. 30 (2022), no. 3, 499–535.CrossRefGoogle 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
Tabuada, G., Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories , C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 15–19.CrossRefGoogle Scholar