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ENHANCED FINITE TRIANGULATED CATEGORIES

Part of: Homological algebra Abelian categories Homological methods Operations and obstructions

Published online by Cambridge University Press:  18 June 2020

Fernando Muro Open the ORCID record for Fernando Muro [Opens in a new window]
Fernando Muro*
Affiliation:
Universidad de Sevilla, Facultad de Matemáticas, Departamento de Álgebra, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain (fmuro@us.es) URL: https://personal.us.es/fmuro
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Abstract

We give a necessary and sufficient condition for the existence of an enhancement of a finite triangulated category. Moreover, we show that enhancements are unique when they exist, up to Morita equivalence.

Keywords

Triangulated categoryA-infinity categoryHochschild cohomologyspectral sequenceobstruction theory

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.) 18G40: Spectral sequences, hypercohomology 55S35: Obstruction theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 3 , May 2022 , pp. 741 - 783
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Amiot, C., On the structure of triangulated categories with finitely many indecomposables, Bull. Soc. Math. France 135(3) (2007), 435–474.CrossRefGoogle Scholar
Auslander, M., Representation theory of artin algebras II, Comm. Algebra 1 (1974), 269–310.CrossRefGoogle Scholar
Auslander, M. and Reiten, I., On a theorem of E. Green on the dual of the transpose, in Representations of Finite-Dimensional Algebras (Tsukuba, 1990), CMS Conf. Proc., Volume 11, pp. 53–65 (American Mathematical Society, Providence, RI, 1991).Google Scholar
Benson, D., Krause, H. and Schwede, S., Realizability of modules over Tate cohomology, Trans. Amer. Math. Soc. 356(9) (2004), 3621–3668.CrossRefGoogle Scholar
Bergh, P. A. and Jorgensen, D. A., Tate-Hochschild homology and cohomology of Frobenius algebras, J. Noncommut. Geom. 7(4) (2013), 907–937.CrossRefGoogle Scholar
Białkowski, J., Deformed preprojective algebras of Dynkin type 𝔼6 , Comm. Algebra 47 (2019), 1568–1577.CrossRefGoogle Scholar
Białkowski, J., Erdmann, K. and Skowroński, A., Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc. 359(6) (2007), 2625–2650.CrossRefGoogle Scholar
Białkowski, J., Erdmann, K. and Skowroński, A., Deformed preprojective algebras of generalized Dynkin type 𝕃n: classification and symmetricity, J. Algebra 345(2011) 150–170.CrossRefGoogle Scholar
Białkowski, J. and Skowroński, A., Nonstandard additively finite triangulated categories of Calabi–Yau dimension one in characteristic 3, Algebra Discrete Math. 6(3) (2007), 27–37.Google Scholar
Błaszkiewicz, M. and Skowroński, A., On self-injective algebras of finite representation type, Colloq. Math. 127(1) (2012), 111–126.CrossRefGoogle Scholar
Bolla, M. L., Isomorphisms between endomorphism rings of progenerators, J. Algebra 87(1) (1984), 261–281.CrossRefGoogle Scholar
Bondal, A. I. and Kapranov, M. M., Enhanced triangulated categories, Mat. USSR Sb. 70 (1991), 93–107.CrossRefGoogle Scholar
Bousfield, A. K. and Kan, D. M., Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, Volume 304 (Springer, Berlin-New York, 1972).CrossRefGoogle Scholar
Canonaco, A. and Stellari, P., Uniqueness of dg enhancements for the derived category of a Grothendieck category, J. Eur. Math. Soc. (JEMS) 20(11) (2018), 2607–2641.CrossRefGoogle Scholar
Chen, J., Surjections of unit groups and semi-inverses, J. Commut. Algebra (2019), https://projecteuclid.org/euclid.jca/1545015624.Google Scholar
Dwyer, W. G. and Kan, D. M., A classification theorem for diagrams of simplicial sets, Topology 23 (1984), 139–155.CrossRefGoogle Scholar
Erdmann, K. and Holm, T., Twisted bimodules and Hochschild cohomology for self-injective algebras of class A n , Forum Math. 11(2) (1999), 177–201.Google Scholar
Erdmann, K. and Skowroński, A., Periodic algebras, in Trends in Representation Theory of Algebras and Related Topics, EMS Ser. Congr. Rep., pp. 201–251 (European Mathematical Society, Zürich, 2008).CrossRefGoogle Scholar
Eu, C.-H. and Schedler, T., Calabi–Yau Frobenius algebras, J. Algebra 321(3) (2009), 774–815.CrossRefGoogle Scholar
Franke, J., Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence, K-theory 139 (1996).Google Scholar
Freyd, P., Stable homotopy, in Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), pp. 121–172 (Springer, New York, 1966).CrossRefGoogle Scholar
Hanihara, N., Auslander correspondence for triangulated categories, arXiv:1805.07585 [math], 2018.Google Scholar
Holm, T., Hochschild cohomology of Brauer tree algebras, Comm. Algebra 26(11) (1998), 3625–3646.CrossRefGoogle Scholar
Kadeishvili, T. V., On the theory of homology of fiber spaces, Uspekhi Mat. Nauk 35(3(213)) (1980), 183–188. International Topology Conference (Moscow State Univ., Moscow, 1979).Google Scholar
Kadeishvili, T. V., The algebraic structure in the homology of an A (∞)-algebra, Soobshcheniya Akademii Nauk Gruzinskoĭ SSR 108(2) (1982), 249–252. 1983.Google Scholar
Kajiura, H., On A ∞-enhancements for triangulated categories, J. Pure Appl. Algebra 217(8) (2013), 1476–1503.CrossRefGoogle Scholar
Keller, B., Introduction to A ∞-infinity algebras and modules, Homology Homotopy Appl. 3 (2001), 1–35.CrossRefGoogle Scholar
Keller, B., A remark on a theorem by Claire Amiot, C. R. Math. Acad. Sci. Paris 356 (2018), 984–986.CrossRefGoogle Scholar
Krause, H., Report on locally finite triangulated categories, J. K-Theory 9(3) (2012), 421–458.CrossRefGoogle Scholar
Lam, T. Y., Lectures on Modules and Rings, Graduate Texts in Mathematics, Volume 189 (Springer, New York, 1999).CrossRefGoogle Scholar
Lefèvre-Hasegawa, K., Sur les $A_{\infty }$ -catégories, PhD thesis, Université Paris 7 (2003).Google Scholar
Lunts, V. A. and Orlov, D. O., Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23(3) (2010), 853–908.CrossRefGoogle Scholar
Mitchell, B., Rings with several objects, Adv. Math. 8 (1972), 1–161.CrossRefGoogle Scholar
Muro, F., On the functoriality of cohomology of categories, J. Pure Appl. Algebra 204(3) (2006), 455–472.CrossRefGoogle Scholar
Muro, F., Moduli spaces of algebras over nonsymmetric operads, Algebr. Geom. Topol. 14(3) (2014), 1489–1539.CrossRefGoogle Scholar
Muro, F., Cylinders for non-symmetric DG-operads via homological perturbation theory, J. Pure Appl. Algebra 220(9) (2016), 3248–3281.CrossRefGoogle Scholar
Muro, F., Homotopy units in A-infinity algebras, Trans. Amer. Math. Soc. 368(3) (2016), 2145–2184.CrossRefGoogle Scholar
Muro, F., The first obstructions to enhancing a triangulated category, Math. Z. (2019), https://doi.org/10.1007/s00209-019-02438-y.CrossRefGoogle Scholar
Muro, F., Enhanced A ∞-obstruction theory, J. Homotopy Relat. Struct. 15 (2020), 61–112.CrossRefGoogle Scholar
Muro, F. and Roitzheim, C., Homotopy theory of bicomplexes, J. Pure Appl. Algebra 223(5) (2019), 1913–1939.CrossRefGoogle Scholar
Patchkoria, I., On exotic equivalences and a theorem of Franke, Bull. Lond. Math. Soc. 49 (2017), 1085–1099.CrossRefGoogle Scholar
Pstrągowski, P., Chromatic homotopy is algebraic when $p>n^{2}+n+1$ , arXiv:1810.12250 [math] (2018).n^{2}+n+1$+,+arXiv:1810.12250+[math]+(2018).>Google Scholar
Puppe, D., On the formal structure of stable homotopy theory, in Colloquium on Algebraic Topology, pp. 65–71 (Matematisk Institut, Aarhus Universitet, Aarhus, 1962).Google Scholar
Riedtmann, C., Algebren, Darstellungsköcher, überlagerungen und zurück, Comment. Math. Helv. 55 (1980), 199–224.CrossRefGoogle Scholar
Riedtmann, C., Representation-finite self-injective algebras of class A n , in Representation Theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Mathematics, Volume 832, pp. 449–520 (Springer, Berlin, 1980).Google Scholar
Riedtmann, C., Representation-finite self-injective algebras of class D n , Compos. Math. 49(2) (1983), 231–282.Google Scholar
Rizzardo, A. and Van den Bergh, M., A note on non-unique enhancements, Proc. Amer. Math. Soc. 147 (2019), 451–453.CrossRefGoogle Scholar
Schlichting, M., A note on K-theory and triangulated categories, Invent. Math. 150(1) (2002), 111–116.CrossRefGoogle Scholar
Schwede, S., The stable homotopy category has a unique model at the prime 2, Adv. Math. 164(1) (2001), 24–40.CrossRefGoogle Scholar
Schwede, S., The stable homotopy category is rigid, Ann. of Math. (2) 166(3) (2007), 837–863.CrossRefGoogle Scholar
Schwede, S. and Shipley, B., Algebras and modules in monoidal model categories, Proc. Lond. Math. Soc. (3) 80(2) (2000), 491–511.CrossRefGoogle Scholar
Schwede, S. and Shipley, B., A uniqueness theorem for stable homotopy theory, Math. Z. 239(4) (2002), 803–828.CrossRefGoogle Scholar
Skowroński, A., Selfinjective algebras: finite and tame type, in Trends in Representation Theory of Algebras and Related Topics, Contemp. Math., Volume 406, pp. 169–238 (American Mathematical Society, Providence, RI, 2006).CrossRefGoogle Scholar
Skowroński, A. and Yamagata, K., Frobenius Algebras. I, EMS Textbooks in Mathematics, (European Mathematical Society (EMS), Zürich, 2011).CrossRefGoogle Scholar
Street, R., Homotopy classification of filtered complexes, PhD thesis, University of Sydney (1969).Google Scholar
Suarez-Alvarez, M., The Hilton–Heckmann argument for the anti-commutativity of cup products, Proc. Amer. Math. Soc. 132(8) (2004), 2241–2246.CrossRefGoogle Scholar
Tabuada, G., Invariants additifs de DG-catégories, Int. Math. Res. Not. IMRN 2005(53) (2005), 3309–3339.CrossRefGoogle Scholar
Tabuada, G., Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories, C. R. Math. Acad. Sci. Paris 340(1) (2005), 15–19.CrossRefGoogle Scholar
Tabuada, G., Addendum à Invariants Additifs de dg-Catégories, Int. Math. Res. Not. IMRN 2006 (2006), Art. ID 75853, 3.Google Scholar
Tabuada, G., Corrections à Invariants Additifs de DG-catégories, Int. Math. Res. Not. IMRN 2007(24) (2007), Art. ID rnm149, 17.Google Scholar
Toën, B. and Vezzosi, G., Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193(902) (2008), x+224.Google Scholar
Xiao, J. and Zhu, B., Locally finite triangulated categories, J. Algebra 290(2) (2005), 473–490.CrossRefGoogle Scholar
Yoshino, Y., Cohen–Macaulay Modules Over Cohen–Macaulay Rings, London Mathematical Society Lecture Note Series, Volume 146 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar