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Dominant Auslander–Gorenstein algebras and mixed cluster tilting

  • Aaron Chan [1] ; Osamu Iyama [2] ; René Marczinzik [3]
    1. [1] Nagoya University

      Nagoya University

      Naka-ku, Japón

    2. [2] University of Tokyo

      University of Tokyo

      Japón

    3. [3] University of Bonn

      University of Bonn

      Kreisfreie Stadt Bonn, Alemania

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  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 32, Nº. 3, 2026
  • Idioma: inglés
  • DOI: 10.1007/s00029-026-01126-0
  • Enlaces
  • Resumen

    • We introduce the class of dominant Auslander–Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander–Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander–Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra T (A) of a d-representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well-known result that d-representationfinite algebras are fractionally Calabi–Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander–Gorenstein algebra admit mixed precluster tilting modules.

  • Referencias bibliográficas
    • Adachi, T., Tsukamoto, M.: Tilting modules and dominant dimension with respect to injective modules. Q. J. Math. 72(3), 855–884 (2021)
    • Artin, M., Schelter, W.F.: Graded algebras of global dimension 3. Adv. Math. 66(2), 171–216 (1987)
    • Assem, I., Simson, D., Skowronski, A.: Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation...
    • Auslander, M.: Representation Dimension of Artin Algebras. Queen Mary College Mathematics Notes, (1971)
    • Auslander, M., Bridger, M.: Stable module theory, Memoirs of the American Mathematical Society, No. 94 American Mathematical Society, Providence,...
    • Auslander, M., Platzeck, M., Todorov, G.: Homological theory of idempotent Ideals. Trans. Am. Math. Soc. 332(2), 667–692 (1992)
    • Auslander, M., Reiten, I.: k-Gorenstein algebras and syzygy modules. J. Pure Appl. Alg. 92(1), 1–27 (1994)
    • Auslander, M., Reiten, I., Smalø, S.: Representation Theory of Artin Algebras Cambridge Studies in Advanced Mathematics, 36. Cambridge University...
    • Auslander, M., Solberg, Ø.: Gorenstein algebras and algebras with dominant dimension at least 2. Comm. Alg. 21(11), 3897–3934 (1993)
    • Björk, J.: Analytic D-modules and applications. Mathematics and its Applications (Dordrecht). 247. Dordrecht: Kluwer Academic Publishers....
    • Bongartz, K.: On representation-finite selfinjective algebras, coverings, multiplicative bases ... arXiv:2305.17418
    • Bretscher, O., Läser, C., Riedtmann, C.: Self-injective and simply connected algebras. Manuscripta Math. 36(3), 253–307 (1981/82)
    • Cartan, H., Eilenberg, S.: Homological algebra. Princeton Landmarks in Mathematics. Princeton University Press (1999)
    • Chan, A., Darpö, E., Iyama, O., Marczinzik, R.: Periodic trivial extension algebras and fractionally Calabi-Yau algebras. Ann. Sci. Éc. Norm....
    • Chan, A., Iyama, O., Marczinzik, R.: Auslander-Gorenstein algebras from Serre-formal algebras via replication. Adv. Math. 345, 222–262 (2019)
    • Chan, A., Iyama, O., Marczinzik, R. :Auslander-regular algebras and Koszul duality, in preparation
    • Chan, A., Marczinzik, R.: On representation-finite gendo-symmetric biserial algebras. Algebr. Represent. Theory 22(1), 141–176 (2019)
    • Chen, H., Koenig, S.: Ortho-symmetric modules, Gorenstein algebras and derived equivalences. Int. Math. Res. Not. 2016(22), 6979–7037 (2016)
    • Chen, X.: Gorenstein Homological Algebra of Artin Algebras. arXiv:1712.04587
    • Clark, J.: Auslander–Gorenstein Rings for Beginners. International Symposium on Ring Theory 95-115 (2001)
    • Cruz, T., Marczinzik, R.: On properly stratified Gorenstein algebras. J. Pure Appl. Alg. 225(12), 15 (2021)
    • Dao, H., Iyama, O., Takahashi, R., Wemyss, M.: Gorenstein modifications and -Gorenstein rings. J. Alg. Geom. 29(4), 729–751 (2020)
    • Darpö, E., Iyama, O.: -representation-finite self-injective algebras. Adv. Math. 362, 106932 (2020)
    • Darpö, E., Kringeland, T.: Classification of the d-representation-finite trivial extensions of quiver algebras. arXiv:2103.15380
    • Dyckerhoff, T., Jasso, G., Lekili, Y.: The symplectic geometry of higher Auslander algebras: symmetric products of disks. Forum Math. Sigma...
    • Dyckerhoff, T., Jasso, G., Walde, T.: Simplicial structures in higher Auslander-Reiten theory. Adv. Math. 355, 106762 (2019)
    • Enomoto, H.: Classifications of exact structures and Cohen-Macaulay-finite algebras. Adv. Math. 335, 838–877 (2018)
    • Erdmann, K., Holm, T.: Maximal n-orthogonal modules for selfinjective algebras. Proc. Amer. Math. Soc. 136, 3069–3078 (2008)
    • Erdmann, K., Holm, T., Iyama, O., Schröer, J.: Radical embeddings and representation dimension. Adv. Math. 185(1), 159–177 (2004)
    • Fang, M., Koenig, S.: Gendo-symmetric algebras, canonical comultiplication, bar cocomplex and dominant dimension. Trans. Amer. Math. Soc....
    • Fomin, S., Zelevinsky, A.: Cluster algebras I. Foundations. J. Amer. Math. Soc. 15(2), 497–529 (2002)
    • Fossum, R., Griffith, P., Reiten, I.: Trivial extensions of abelian categories. Homological algebra of trivial extensions of abelian categories...
    • Grant, J., Iyama, O.: Higher preprojective algebras, Koszul algebras, and superpotentials. Compos. Math. 156(12), 2588–2627 (2020)
    • Grevstad, J.F.: Higher Auslander–Solberg correspondence for exact categories, arXiv:2207.05547
    • Green, E.: Introduction to Koszul algebras. Representation theory and algebraic geometry (Waltham, MA, 1995), 45–62, London Math. Soc. Lecture...
    • Hanihara, N.: Auslander correspondence for triangulated categories. Alg. Number Theory 14(8), 2037–2058 (2020)
    • Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note...
    • Haugland, J., Sandøy, M.H.: Higher Koszul duality and connections with n-hereditary algebras, arXiv:2101.12743
    • Henrard, R., Kvamme, S., van Roosmalen, A.: Auslander’s formula and correspondence for exact categories. Adv. Math. 401, 108296 (2022)
    • Herschend, M., Iyama, O.: -representation-finite algebras and twisted fractionally Calabi-Yau algebras. Bull. Lond. Math. Soc. 43(3), 449–466...
    • Herschend, M., Iyama, O.: Selfinjective quivers with potential and 2-representation-finite algebras. Compos. Math. 147(6), 1885–1920 (2011)
    • Huang, Z., Zhang, X.: Higher Auslander algebras admitting trivial maximal orthogonal subcategories. J. Alg. 330, 375–387 (2011)
    • Huang, Z., Zhang, X.: The existence of maximal n-orthogonal subcategories. J. Alg. 321(10), 2829–2842 (2009)
    • Huang, Z., Zhang, X.: Trivial maximal 1-orthogonal subcategories for Auslander 1-Gorenstein algebras. J. Aust. Math. Soc. 94(1), 133–144 (2013)
    • Iyama, O.: -categories III: Auslander orders and Auslander-Reiten quivers. Alg. Represent. Theory 8(5), 601–619 (2005)
    • Iyama, O.: Symmetry and duality on -Gorenstein rings. J. Alg. 269(2), 528–535 (2003)
    • Iyama, O.: Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210(1), 22–50 (2007)
    • Iyama, O.: Auslander correspondence. Adv. Math. 210(1), 51–82 (2007)
    • Iyama, O.: Cluster tilting for higher Auslander algebras. Adv. Math. 226(1), 1–61 (2011)
    • Iyama, O., Marczinzik, R.: Distributive lattices and Auslander regular algebras. Adv. Math. 398, 108233 (2022)
    • Iyama, O., Oppermann, S.: -representation-finite algebras and -APR tilting. Trans. Amer. Math. Soc. 363(12), 6575–6614 (2011)
    • Iyama, O., Oppermann, S.: Stable categories of higher preprojective algebras. Adv. Math. 244, 23–68 (2013)
    • Iyama, O., Solberg, Ø.: Auslander-Gorenstein algebras and precluster tilting. Adv. Math. 326, 200–240 (2018)
    • Jasso, G., Keller, B., Muro, F.: The Triangulated Auslander–Iyama Correspondence arXiv:2208.14413
    • Jasso, G., Külshammer, J.: Higher Nakayama algebras I: Construction. With an appendix by Külshammer and Chrysostomos Psaroudakis and an appendix...
    • Koenig, S., Slungard, I., Xi, C.: Double centralizer properties, dominant dimension, and tilting modules. J. Alg. 240(1), 393–412 (2001)
    • Ko, H., Mazorchuk, V., Mrden, R.: Some homological properties of category O. VI. Doc. Math. 26, 1237–1269 (2021)
    • Li, S., Marczinzik, R., Zhang, S.: Gorenstein projective dimensions of modules over minimal Auslander-Gorenstein algebras. Alg. Colloq. 28(2),...
    • Li, S., Zhang, S.: Algebras with finite relative dominant dimension and almost -precluster tilting modules. J. Pure Appl. Alg. 225(2), 106498...
    • Ma, B., Sauter, J.: On faithfully balanced modules, F-cotilting and F-Auslander algebras. J. Alg. 556, 1115–1164 (2020)
    • Madsen, D., Marczinzik, R., Zaimi, G.: On the classification of higher Auslander algebras for Nakayama algebras. J. Alg. 556, 776–805 (2020)
    • Marczinzik, R.: Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra. arXiv:1608.04212
    • Marczinzik, R., Rubey, M., Stump, C.: A combinatorial classification of 2-regular simple modules for Nakayama algebras. J. Pure Appl. Alg....
    • Marczinzik, R., Vaso, L.: Existence of a 2-cluster tilting module does not imply finite complexity. J. Alg. 598, 385–391 (2022)
    • Martinez Villa, R.: Modules of dominant and codominant dimension. Comm. Alg. 20(12), 3515–3540 (1992)
    • McMahon, J.: Fabric idempotents and higher Auslander-Reiten theory. J. Pure Appl. Alg. 224(8), 106343 (2020)
    • Mizuno, Y.: A Gabriel-type theorem for cluster tilting. Proc. Lond. Math. Soc. (3) 108(4), 836–868 (2014)
    • Mueller, B.: The classification of algebras by dominant dimension. Can. J. Math. 20, 398–409 (1968)
    • Nguyen, V.C., Reiten, I., Todorov, G., Zhu, S.: Dominant dimension and tilting modules. Math. Z. 292, 947–973 (2019)
    • Pasquali, A.: Self-injective Jacobian algebras from Postnikov diagrams. Alg. Represent. Theory 23(3), 1197–1235 (2020)
    • Pressland, M., Sauter, J.: Special tilting modules for algebras with positive dominant dimension. Glasg. Math. J. 64(1), 79–105 (2022)
    • The QPA-team, QPA - Quivers, path algebras and representations - a GAP package, Version 1.25, 2016, https://folk.ntnu.no/oyvinso/QPA/
    • Riedtmann, C.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55, 199–224 (1980)
    • Ringel, C.M.: Linear Nakayama algebras which are higher Auslander algebras. Comm. Alg. 50(11), 4842–4881 (2022)
    • Sandøy, M.H., Thibault, L.: Classification results for -hereditary monomial algebras arXiv:2101.12746
    • Sen, E.: Higher Auslander Algebras arising from Dynkin Quivers and -Representation Finite Algebras arXiv:2307.13262
    • Skowronski, A., Yamagata, K.: Frobenius Algebras I: Basic Representation Theory. EMS Textbooks in Mathematics (2011)
    • Skowronski, A., Yamagata, K.: Frobenius Algebras II: Tilted and Hochschild extension algebras. EMS Textbooks in Mathematics (2017)
    • Tachikawa, H.: Quasi-Frobenius Rings and Generalizations: QF-3 and QF-1 Rings (Lecture Notes in Mathematics 351) Springer (1973)
    • van den Bergh, M.: Non-commutative crepant resolutions. The legacy of Niels Henrik Abel, 749–770, Springer, Berlin (2004)
    • Vaso, L.: -cluster tilting subcategories of representation-directed algebras. J. Pure Appl. Alg. 223(5), 2101–2122 (2019)
    • Van Oystaeyen, F.: Algebraic geometry for associative algebras. Pure and applied Mathematics. Marcel Dekker Inc., (2000)
    • Williams, N.: New interpretations of the higher Stasheff-Tamari orders. Adv. Math. 407, 108552 (2022)
    • Wu, Y.: Relative cluster categories and Higgs categories. arXiv:2109.03707
    • Yamagata, K.: Frobenius Algebras in Hazewinkel, M. (editor): Handbook of Algebra, North-Holland, Amsterdam, Volume I, 841-887, (1996)
    • Yekutieli, A.: Derived categories. Cambridge Studies in Advanced Mathematics, 183. Cambridge University Press, Cambridge, (2020)
    • Zito, S.: 1-Auslander-Gorenstein algebras which are tilted. J. Alg. 555, 265–274 (2020)
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