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On Frobenius (Completed) Orbit Categories

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Abstract

Let 𝓔 be a Frobenius category, \({\mathcal P}\) its subcategory of projective objects and F : 𝓔 → 𝓔 an exact automorphism. We prove that there is a fully faithful functor from the orbit category 𝓔/F into \(\operatorname {gpr}({\mathcal P}/F)\), the category of finitely-generated Gorenstein-projective modules over \({\mathcal P}/F\). We give sufficient conditions to ensure that the essential image of 𝓔/F is an extension-closed subcategory of \(\operatorname {gpr}({\mathcal P}/F)\). If 𝓔 is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category \({\mathcal E} \ \widehat {\!\! /} F\) is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.

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References

  1. Asashiba, H.: A generalization of Gabriel’s Galois covering functors II: 2-categorical Cohen-Montgomery duality. H Appl. Categor. Struct (2015)

  2. Asashiba, H.: A generalization of Gabriel’s Galois covering functors and derived equivalences. J. Algebra 334(1), 109–149 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86, 111–152 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bondal, A.I., Kapranov, M.M.: Enhanced triangulated categories. Mat. Sb 181(5), 669–683 (1990). translation in Math. USSR-Sb. 70 no. 1, 93–107

    MATH  Google Scholar 

  5. Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi-Yau categories and unipotent groups. Amer. J. Math. 133(4), 835–887 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buan, A.B., Marsh, R.J., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bühler, T.: Exact categories. Expo. Math. 28, 1–69 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, X.U.: Three results on Frobenius categories. Mathematische Zeitschrift 270, 43–58 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cibils, C., Marcos, E.: Skew category Galois covering and smash product of a k-category. Proc. AMS 134(1), 39–50 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Drinfeld, V.: DG Quotients of DG categories. J. Algebra 272(2), 643–691 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Amer. Math. Soc. 15, 497–529 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fu, C.h.: On root categories of finite-dimensional algebras. J. Algebra 370, 233–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fu, C.h.: Auslander-Reiten sequences and representation-finite algebras, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont.), pp 1–71. Springer, Berlin (1980)

    Google Scholar 

  14. Happel, D.: On the derived category of a finite-dimesional algebra. Comment Math. Helv. 62(3), 339–389 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. Cambridge university press cambridge (1988)

  16. Igusa, K., Todorov, G.: Continuous cluster categories I, algebras and representation theory. doi:10.1007/s10468-014-9481-z (2014)

  17. Keller, B.: Chain complexes and stable categories. Manus. Math. 67, 379–417 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Keller, B.: Deriving DG categories. Ann. Scient. Ec. Norm. Sup. 27, 63–102 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Keller, B.: On differential graded categories. In: International Congress of Mathematicians, vol. II. European Mathematical Society, Zürich, pp. 151–190. MR2275593 (2008g:18015) (2006)

  20. Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551–581 (2005)

    MathSciNet  MATH  Google Scholar 

  21. Keller, B.: Corrections to ”On triangulated orbit categories”, available at the author’s webpage

  22. Keller, B.: Cluster algebras, quiver representations and triangulated categories Workshop on Triangulated Categories (Leeds) (2006)

  23. Keller, B., Scherotzke, S.: Graded quiver varieties and derived categories. J. Reine Angew. Math., doi:10.1515/crelle-2013-0124

  24. Keller, B., Vossieck, D.: Sous les catégories dérivées. C. R. Acad. Sci. Paris 305, 225–228 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Krause, H.: Functors on locally finitely presented additive categories. Colloq. Math. 75(1), 105–132 (1998)

    MathSciNet  MATH  Google Scholar 

  26. Nájera Chávez, A.: A 2-Calabi-Yau realization of finite-type cluster algebras with universal coefficients. arXiv:1512.07939

  27. Qin, F.: Quantum groups via cyclic quiver varieties I. Compos. Math. 152(2), 299–326 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Quillen, D.: Higher algebraic K-theory I. Springer LNM 341, 85–147 (1973)

    MathSciNet  MATH  Google Scholar 

  29. Riedtmann, C.h.: Algebren, darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55 2, 199–224 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Riedtmann, C.h.: Representation-finite self-injective algebras of class A n , Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832, pp 449–520. Springer, Berlin (1980)

    Google Scholar 

  31. Scherotzke, S.: Quiver varieties and self-injective algebras, arXiv:1405.4729

  32. Tabuada, G.: Invariants additifs de dg-catégories. Int. Math. Res. Notices 53, 3309–3339 (2005)

    Article  MATH  Google Scholar 

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  1. Instituto de Matemáticas Representación Oaxaca, Universidad Nacional Autónoma de México, León 2, altos, col. Centro, C.P. 68000, Oaxaca de Juárez, Oaxaca, Mexico

    Alfredo Nájera Chávez

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  1. Alfredo Nájera Chávez
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Correspondence to Alfredo Nájera Chávez.

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Nájera Chávez, A. On Frobenius (Completed) Orbit Categories. Algebr Represent Theor 20, 1007–1027 (2017). https://doi.org/10.1007/s10468-017-9672-5

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