Abstract
Given \(n\leq d<\infty \), we investigate the existence of algebras of global dimension d which admit an n-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras A and B, a projective A-module P and an injective B-module I satisfying certain conditions, we show how we can construct a new representation-directed algebra in such a way that the representation theory of Λ is completely described by the representation theories of A and B. Next we introduce n-fractured subcategories which generalize n-cluster tilting subcategories for representation-directed algebras. We then show how one can construct an n-cluster tilting subcategory for Λ by using n-fractured subcategories of A and B. As an application of our construction, we show that if n is odd and d ≥ n then there exists an algebra admitting an n-cluster tilting subcategory and having global dimension d. We show the same result if n is even and d is odd or d ≥ 2n.
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Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory. Elements of the Representation Theory of Associative Algebras. Cambridge University Press. https://books.google.se/books?id=ayNHpi3tYhQC (2006)
Auslander, M., Reiten, I., SmalØ, O.: Representation theory of Artin algebras Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)
Auslander, M., Smalø, O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)
Bekkert, V., Coelho, F.U., Wagner, H.: Tree oriented pullback. Commun. Algebra 43(10), 4247–4257 (2015). https://doi.org/10.1080/00927872.2014.942422
Darpö, E., Iyama, O.: d-representation-finite self-injective algebras. Adv. Math. 362, 106932 (2020). https://doi.org/10.1016/j.aim.2019.106932. http://www.sciencedirect.com/science/article/pii/S000187081930547X
Erdmann, K., Holm, T.: Maximal n-orthogonal modules for selfinjective algebras. Proc. Amer. Math. Soc. 136(9), 3069–3078 (2008). https://doi.org/10.1090/S0002-9939-08-09297-6. http://www.ams.org/journals/proc/2008-136-09/S0002-9939-08-09297-6
Happel, D., Ringel, C.M.: Construction of Tilted Algebras. In: Auslander, M., Lluis, E. (eds.) Representations of Algebras, pp 125–144. Springer, Berlin (1981)
Herschend, M., Iyama, O.: n-representation-finite algebras and twisted fractionally Calabi-Yau algebras. Bull. Lond. Math. Soc. 43(3), 449–466 (2010). https://doi.org/10.1112/blms/bdq101. arXiv:0908.3510v2
Herschend, M., Iyama, O.: Selfinjective quivers with potential and 2-representation-finite algebras. Compos. Math. 147(6), 1885–1920 (2010). https://doi.org/10.1112/S0010437X11005367. arXiv:1006.1917v2
Herschend, M., Iyama, O., Oppermann, S.: n-representation infinite algebras. Adv. Math. 252, 292–342 (2014). https://doi.org/10.1016/j.aim.2013.09.023. http://www.sciencedirect.com/science/article/pii/S0001870813003642
Igusa, K., Platzeck, M., Todorov, G., Zacharia, D.: Auslander algebras of finite representation type. Commun. Algebra 15(1-2), 377–424 (1987). https://doi.org/10.1080/00927878708823424
Iyama, O.: Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories. Adv. Math. 210(1), 22–50 (2007). https://doi.org/10.1016/j.aim.2006.06.002. http://www.sciencedirect.com/science/article/pii/S0001870806001721
Iyama, O.: Auslander-Reiten Theory Revisited. In: Trends in Representation Theory of Algebras and Related Topics, EMS Ser. Congr. Rep. https://doi.org/10.4171/062-1/8, pp 349–397. Eur. Math. Soc., Zürich (2008)
Iyama, O., Oppermann, S.: n-representation-finite algebras and n-APR tilting. Trans. Amer. Math. Soc. 363(12), 6575–6614 (2010). arXiv:0909.0593v2
Iyama, O., Oppermann, S.: Stable categories of higher preprojective algebras. Adv. Math. 244, 23–68 (2013). https://doi.org/10.1016/j.aim.2013.03.013
Jasso, G.: n-Abelian and n-exact categories. Math. Z. 283(3), 703–759 (2016). https://doi.org/10.1007/s00209-016-1619-8
Jasso, G., Külshammer, J., Psaroudakis, C., Kvamme, S.: Higher Nakayama algebras I: Construction. Adv. Math. 351, 1139 – 1200 (2019). https://doi.org/10.1016/j.aim.2019.05.026
Kupisch, H.: Beiträge Zur Theorie Nichthalbeinfacher Ringe Mit Minimalbedingung. Ph.D. thesis, NA Heidelberg (1958)
Lévesque, J.: Nakayama oriented pullbacks and stably hereditary algebras. J. Pure Appl. Algebra 212 (5), 1149 – 1161 (2008). https://doi.org/10.1016/j.jpaa.2007.08.001. http://www.sciencedirect.com/science/article/pii/S0022404907002186
MacLane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1998)
Milnor, J.: Introduction to Algebraic K-Theory. (AM-72). Princeton University Press. http://www.jstor.org/stable/j.ctt1b9x0xv (1971)
Ringel, C.M.: Representation theory of Dynkin quivers. Three contributions. Front. Math. China 11(4), 765–814 (2016). https://doi.org/10.1007/s11464-016-0548-5
Vaso, L.: n-Cluster tilting subcategories of representation-directed algebras. Journal of Pure and Applied Algebra. https://doi.org/10.1016/j.jpaa.2018.07.010. http://www.sciencedirect.com/science/article/pii/S0022404918301841 (2018)
Acknowledgements
The author wishes to thank his advisor Martin Herschend for the constant support and suggestions during the preparation of this article. He also wishes to thank Andrea Pasquali for offering helpful suggestions about the manuscript. Finally he wishes to thank an anonymous referee for their careful reading of the article as well as their many helpful and detailed comments and corrections which improved the presentation considerably.
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Vaso, L. Gluing of n-Cluster Tilting Subcategories for Representation-directed Algebras. Algebr Represent Theor 24, 715–781 (2021). https://doi.org/10.1007/s10468-020-09967-9
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DOI: https://doi.org/10.1007/s10468-020-09967-9