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Resolutions of mesh algebras: periodicity and Calabi-Yau dimensions. (English) Zbl 1272.16017

Summary: A triangulated category \((\mathcal T,\Sigma)\) is said to be Calabi-Yau of dimension \(d\) if \(\Sigma^d\) is a Serre functor. We determine which stable module categories of self-injective algebras \(\Lambda\) of finite type are Calabi-Yau and compute their Calabi-Yau dimensions, correcting errors in previous work. We first show that the Calabi-Yau property of \(\underline{\text{mod}}\)-\(\Lambda\) can be detected in the minimal projective resolution of the stable Auslander algebra \(\Gamma\) of \(\Lambda\), over its enveloping algebra. We then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the (generalized) preprojective algebras of Dynkin graphs. For representation-finite self-injective algebras of torsion order \(t=1\), we obtain a complete description of their stable Calabi-Yau properties, but only partial results for those algebras of torsion order \(t=2\). We also obtain some new information about the periods of the representation-finite self-injective algebras of torsion order \(t>1\). Finally, we describe how these questions can also be approached by realizing the stable categories of representation-finite self-injective algebras as orbit categories of the bounded derived categories of hereditary algebras, and illustrate this technique with several explicit computations that our previous methods left unsettled.

MSC:

16G20 Representations of quivers and partially ordered sets
16E05 Syzygies, resolutions, complexes in associative algebras
16E35 Derived categories and associative algebras
18E30 Derived categories, triangulated categories (MSC2010)
16D90 Module categories in associative algebras
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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[1] Amiot C.: On the structure of triangulated categories with finitely many indecomposables. Bull. Soc. Math. France 135(3), 435–474 (2007) · Zbl 1158.18005
[2] Asashiba H.: The derived equivalence classification of representation-finite selfinjective algebras. J. Algebra 214(1), 182–221 (1999) · Zbl 0949.16013 · doi:10.1006/jabr.1998.7706
[3] Asashiba H.: On a lift of an individual stable equivalence to a standard derived equivalence for representation-finite self-injective algebras. Algebr. Represent. Theory 6(4), 427–447 (2003) · Zbl 1062.16023 · doi:10.1023/B:ALGE.0000003543.02426.4d
[4] Asashiba H.: A generalization of Gabriel’s Galois covering functors and derived equivalences. J. Algebra 334, 109–149 (2011) · Zbl 1241.18002 · doi:10.1016/j.jalgebra.2011.03.002
[5] Auslander, M., Reiten, I.: On a theorem of E. Green on the dual of the transpose. Proc. ICRA V, CMS Conf. Proc. 11, 53–65 (1991) · Zbl 0754.16008
[6] Białkowski J., Erdmann K., Skowroński A.: Deformed preprojective algebras of generalized Dynkin type. Trans. Am. Math. Soc. 359(6), 2625–2650 (2007) · Zbl 1117.16005 · doi:10.1090/S0002-9947-07-03948-7
[7] Białkowski, J., Erdmann, K., Skowroński, A.: Deformed mesh algebras of generalized Dynkin type. In preparation · Zbl 1117.16005
[8] Białkowski J., Skowroński A.: Calabi-Yau stable module categories of finite type. Colloq. Math. 109(2), 257–269 (2007) · Zbl 1168.16003 · doi:10.4064/cm109-2-8
[9] Bocklandt R.: Graded Calabi-Yau algebras of dimension 3 (Appendix by M. Van den Bergh). J. Pure Appl. Algebra 212(1), 14–32 (2008) · Zbl 1132.16017 · doi:10.1016/j.jpaa.2007.03.009
[10] Brenner S., Butler M.C.R., King A.D.: Periodic algebras which are almost Koszul. Algebras Represent. Theory 5, 331–367 (2002) · Zbl 1056.16003 · doi:10.1023/A:1020146502185
[11] Buan A.B., Marsh R., Reineke M., Reiten I., Todorov G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006) · Zbl 1127.16011 · doi:10.1016/j.aim.2005.06.003
[12] Buchweitz, R.O.: Finite representation type and periodic Hochschild (co-)homology. In: Trends in the Representation Theory of Finite-Dimensional Algebras (Seattle, WA, 1997), pp. 81–109. Contemp. Math., vol. 229. Amer. Math. Soc., Providence (1998) · Zbl 0934.16009
[13] Cibils C., Marcos E.: Skew category, Galois covering and smash product of a k-category. Proc. Am. Math. Soc. 134(1), 39–50 (2006) · Zbl 1098.18002 · doi:10.1090/S0002-9939-05-07955-4
[14] Cohen M., Montgomery S.: Group-graded rings, smash products, and group actions. Trans. Am. Math. Soc. 282(1), 237–258 (1984) · Zbl 0533.16001 · doi:10.1090/S0002-9947-1984-0728711-4
[15] Dieterich, E.: The Auslander-Reiten quiver of an isolated singularity. Singularities, Representation of Algebras, and Vector Bundles (Lambrecht, 1985), pp. 244–264. Lecture Notes in Math., vol. 1273. Springer, Berlin (1987)
[16] Dieterich E., Wiedemann A.: The Auslander Reiten quiver of a simple curve singularity. Trans. Am. Math. Soc. 294, 455–475 (1986) · Zbl 0603.14019 · doi:10.1090/S0002-9947-1986-0825715-X
[17] Dugas, A.: Periodic resolutions and self-injective algebras of finite type. J. Pure Appl. Algebra 214, 990–1000 (2010). arXiv:0808.1311v2 · Zbl 1214.16006
[18] Dugas, A.: Periodicity of d-cluster-tilted algebras. Preprint (2010). arXiv:1007.2811v1 · Zbl 1279.16007
[19] Erdmann K., Skowroński A.: The stable Calabi-Yau dimension of tame symmetric algebras. J. Math. Soc. Jpn. 58, 97–128 (2006) · Zbl 1167.16013 · doi:10.2969/jmsj/1145287095
[20] Erdmann, K., Skowroński, A.: Periodic algebras. Trends in Representation Theory and Related Topics. European Math. Soc. Series of Congress Reports, pp. 201–251. European Math. Soc. Publ. House, Zurich (2008) · Zbl 1210.16012
[21] Erdmann, K., Snashall, N.: On Hochschild cohomology of preprojective algebras. I, II. J. Algebra 205(2), 391–412, 413–434 (1998) · Zbl 0937.16012
[22] Eu C.-H., Schedler T.: Calabi-Yau Frobenius algebras. J. Algebra 321(3), 774–815 (2009) · Zbl 1230.16009 · doi:10.1016/j.jalgebra.2008.11.003
[23] Green E.L.: Graphs with relations, coverings and group-graded algebras. Trans. Am. Math. Soc. 279(1), 297–310 (1983) · Zbl 0536.16001 · doi:10.1090/S0002-9947-1983-0704617-0
[24] Happel D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62(3), 339–389 (1987) · Zbl 0626.16008 · doi:10.1007/BF02564452
[25] Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. In: London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988) · Zbl 0635.16017
[26] Happel, D.: Hochschild cohomology of finite-dimensional algebras. Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988), pp. 108–126. Lecture Notes in Math., vol. 1404. Springer, Berlin (1989)
[27] Happel, D., Preiser, U., Ringel, C.M.: Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules. Representation Theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), pp. 280–294. Lecture Notes in Math., vol. 832. Springer, Berlin (1980) · Zbl 0446.16032
[28] Happel D., Preiser U., Ringel C.M.: Binary polyhedral groups and Euclidean diagrams. Manuscr. Math. 31(1–3), 317–329 (1980) · Zbl 0436.20005 · doi:10.1007/BF01303280
[29] Holm, T., Jørgensen, P.: Cluster categories, self-injective algebras and stable Calabi-Yau dimensions: type A. Preprint (2008). arXiv:math/0610728v2
[30] Holm, T., Jørgensen, P.: Cluster categories, self-injective algebras and stable Calabi-Yau dimensions: types D and E. Preprint (2008). arXiv:math/0612451v2
[31] Keller B.: On triangulated orbit categories. Documenta Math. 10, 551–581 (2005) · Zbl 1086.18006
[32] Keller, B.: Corrections to ’On triangulated orbit categories’. http://people.math.jussieu.fr/\(\sim\)keller/publ/index.html · Zbl 1086.18006
[33] Keller, B.: Calabi-Yau triangulated categories. In: Trends in Representation Theory of Algebras and Related Topics, pp. 467–489. European Math. Soc. Series of Congress Reports. European Math. Soc. Publ. House, Zurich (2008) · Zbl 1202.16014
[34] Keller, B.: Correction to ’Calabi-Yau triangulated categories’. http://people.math.jussieu.fr/\(\sim\)keller/publ/index.html · Zbl 1202.16014
[35] Keller, B.: On the Calabi-Yau property for higher cluster categories. Private communication (2009)
[36] Keller B., Reiten I.: Acyclic Calabi-Yau categories (Appendix by Michel Van den Bergh). Compos. Math. 144(5), 1332–1348 (2008) · Zbl 1171.18008 · doi:10.1112/S0010437X08003540
[37] Miyachi J., Yekutieli A.: Derived Picard groups of finite-dimensional hereditary algebras. Compos. Math. 129, 341–368 (2001) · Zbl 0999.16012 · doi:10.1023/A:1012579131516
[38] Reiten I., Van den Bergh M.: Noetherian hereditary abelian categories satisfying Serre duality. J. Am. Math. Soc. 15(2), 295–366 (2002) · Zbl 0991.18009 · doi:10.1090/S0894-0347-02-00387-9
[39] Rickard J.: Derived ategories and stable equivalence. J. Pure Appl. Algebra 61(3), 303–317 (1989) · Zbl 0685.16016 · doi:10.1016/0022-4049(89)90081-9
[40] Riedtmann Ch.: Algebren, Darstellungsköcher, Ueberlagerungen und zurück. Comment. Math. Helvetici 55, 199–224 (1980) · Zbl 0444.16018 · doi:10.1007/BF02566682
[41] Yoshino, Y.: Cohen–Macaulay modules over Cohen–Macaulay rings. In: London Mathematical Society Lecture Note Series, vol. 146, Cambridge University Press, Cambridge (1990) · Zbl 0745.13003
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