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Generalized Tilting Theory

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Abstract

Given small dg categories A and B and a B-A-bimodule T, we give necessary and sufficient conditions for the associated derived functors of Hom and the tensor product to be fully faithful. Special emphasis is put on the case when RHom\(_\mathrm{A}\)(T,?) is fully faithful and preserves compact objects, in which case nice recollements situations appear. It is also shown that, given an algebraic compactly generated triangulated category D, all compactly generated co-smashing triangulated subcategories which contain the compact objects appear as the image of such a RHom\(_\mathrm{A}\)(T,?). The results are then applied to the case when A and B are ordinary algebras, comparing the situation with the well-stablished tilting theory of modules. In this way we recover and extend recent results by Bazzoni–Mantese–Tonolo, Chen-Xi and D. Yang.

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References

  1. Anderson, F.K., Fuller, K.R.: Rings and Categories of Modules, vol. 13. Springer, Berlin (1992)

    MATH  Google Scholar 

  2. Angeleri-Hügel, L., Coelho, F.U.: Infinitely generated tilting modules of finite projective dimension. Forum Math. 13, 239–250 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Angeleri-Hügel, L., Herbera, D., Trlifaj, J.: Tilting modules and Gorenstein rings. Forum Math. 18, 211–229 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Angeleri-Hügel, L., Sánchez, J.: Tilting modules over tame hereditary algebras. Journal für die Reine und Angewandte Mathematik 682, 1–48 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Angeleri-Hügel, L., Trlifaj, J.: Tilting theory and the finitistic dimension conjecture. Trans. Am. Math. Soc. 354, 4345–4358 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory. London Math. Soc. Student Texts, vol. 65. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  7. Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1–12 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge St. Adv. Maths, vol. 36. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  10. Bazzoni, S.: Cotilting modules are pure-injective. Proc. Am. Math. Soc. 131, 3665–3672 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bazzoni, S., Mantese, F., Tonolo, A.: Derived equivalence induced by \(n\)-tilting modules. Proc. Am. Math. Soc. 139(12), 4225–4234 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Beilinson, A. A., Bernstein, J., Deligne, P.: Faisceaux pervers. Astérisque 100 (1982)

  13. Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 188(883) (2007)

  14. Bernstein, I., Gelfand, I.M., Ponomariev, V.A.: Coxeter functors and Gabriel’s theorem. Usp. Mat. Nauk 28, 19–23 (1973)

    MathSciNet  Google Scholar 

  15. Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and reconstructions. Izv. Akad. Nauk SSSR Ser. Math. 53(6) 1183–1205, 1337 (1989)

  16. Bondal, A.I., Van den Bergh, M.: Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3(1), 1–36 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Borngartz, K.: Tilted Algebras. Springer Lecture Notes in Mathematics, vol. 904, pp. 26–38 (1981)

  18. Brenner, S., Butler, M.C.R.: Generalization of the Bernstein–Gelfand–Ponomarev Reflection Functors. Representation Theory II, Springer Lecture Notes in Mathematics, vol. 832, pp. 103–169 (1980)

  19. Buan, A.B., Krause, H.: Cotilting modules over tame hereditary algebras. Pacif. J. Math. 211(1), 41–59 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, H., Xi, C.: Good tilting modules and recollements of derived module categories. Proc. Lond. Math. Soc. 104(5), 959–996 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, H., Xi, C.: Ringel modules and homological subcategories. arXiv:1206.0522v1

  22. Colby, R.R.: A generalization of Morita duality and the tilting theorem. Commun. Algebra 17(7), 1709–1722 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Colby, R.R., Fuller, K.R.: Tilting, cotilting, and serially tilted rings. Commun. Algebra 18, 1585–1615 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Colpi, R., Trlifaj, J.: Tilting modules and tilting torsion theories. J. Algebra 178, 614–634 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Deligne, P.: Cohomologie à supports propres. SGA 4, Exposé XVII, Springer-Verlag. Lecture Notes in Mathematics, vol. 305, pp. 252–480 (1973)

  26. Dixmier, J.: Algèbres Envelopantes. Gauthier-Villars, New York (1974)

    Google Scholar 

  27. Donkin, S.: Tilting modules for algebraic groups and finite dimensional algebras. In: Angeleri-Hügel, L., Happel, D., Krause, H. (eds.) Handbook of Tilting Theory. London Math. Soc. LNS, vol. 332, pp. 215–257. Cambridge University Press, Cambridge (2007)

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gabriel, P., Popescu, N.: Charactérisation des catégories abéliennes avec générateurs et limites inductives exactes. C. R. Acad. Sci. Paris 258, 4188–4190 (1964)

    MathSciNet  MATH  Google Scholar 

  31. Gabriel, P., Zisman, M.: Calculus of Factions and Homotopy Theory, vol. 35. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  32. Gelfand, S.I., Manin, YuI: Methods of Homological Algebra. Springer Monographs in Mathematics, Berlin (1996)

    Book  MATH  Google Scholar 

  33. Geigle, W., Lenzing, H.: A class of weighted projective lines arising in the representation theory of finite dimensional algebras. Springer Lecture Notes in Mathematics 1273, 265–297 (1987)

    Article  MATH  Google Scholar 

  34. Geigle, W., Lenzing, H.: Perpendicular categories with applications to representations and sheaves. J. Algebra 144, 273–343 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. Göbel, R., Trlifaj, J.: Approximations and Endomorphism Algebras. De Gruyter, Berlin (2006)

    Book  MATH  Google Scholar 

  36. Happel, D.: On the derived category of a finite dimensional algebra. Comment. Math. Helvetici 62, 339–389 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  37. Happel, D., Reiten, I., Smalo, S.O.: Piecewise hereditary algebras. Arch Math. 66(3), 182–186 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. Happel, D., Reiten, I., Smalo, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 120(575) (1996)

  39. Happel, D., Ringel, C.M.: Tilted algebras. Trans. Am. Math. Soc. 274, 399–443 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hirschhorn, P.S.: Model Categories and Their Localizations. Mathematical Surveys and Monographs 99, AMS (2003)

  41. Keller, B.: Deriving DG categories. Ann. Scient. Ec. Norm. Sup. 27(1), 63–102 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Keller, B.: Chapter of handbook of algebra. In: Hazewinkel, M. (ed.) Derived Categories and Their Uses, vol. 1. Elsevier, New York (1996)

    Google Scholar 

  43. Keller, B.: On Differential Graded Categories. International Congress of Mathematicians, Eur. Math. Soc., Zürich, vol. 2, pp. 151–190 (2006)

  44. Keller, B.: Cluster algebras and cluster categories. Notes from introductory survey lectures given at the IPM. Bull. Iranian Math. Soc. 37(2), part 2, 187–234 (2011)

  45. Keller, B.: Cluster algebras and derived categories. To Appear in the Proceedings of the GCOE Conference Derived categories 2011 Tokyo. http://www.math.jussieu.fr/~keller/publ/index.html

  46. Keller, B., Nicolás, P.: Weight structures and simple dg modules for positive dg algebras. Int. Math. Res. Not. 5, 1028–1078 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Krause, H.: A brown representability theorem via coherent functors. Topology 41, 853–861 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. Krause, H., Solberg, O.: Filtering modules of finite projective dimension. Forum Math. 15(3), 631–650 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lectures on modules and rings, Springer-Verlag Graduate Texts in Mathematics, vol. 189 (1999)

  50. Lenzing, H.: Hereditary categories. In: Angeleri-Hügel, L., Happel, D., Krause, H. (eds.) Handbook of Tilting Theory. London Math. Soc. LNS 332. Cambridge University Press, Cambridge (2007)

  51. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Wiley, New York (1987)

    MATH  Google Scholar 

  52. Mac Lane, S.: Categories for the Working Mathematician, vol. 5. Springer-Verlag Graduate Texts in Mathematics 5

  53. Meltzer, H.: Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines. Mem. Am. Math. Soc. 171 (2004)

  54. Miyashita, Y.: Tilting modules of finite projective dimension. Math. Zeitschr. 193, 113–146 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  55. Neeman, A.: Triangulated Categories Annals of Mathematics studies. Princeton University Press, Princeton (2001)

    Google Scholar 

  56. Nicolás, P.: On Torsion Torsionfree Triples. Ph.D. thesis, Universidad de Murcia (2007)

  57. Nicolás, P., Saorín, M.: Parametrizing recollement data for triangulated categories. J. Algebra 322, 1220–1250 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. Nicolás, P., Saorín, M.: Lifting and restricting recollement. Appl. Categor. Struct. 19, 557–596 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  59. Reiten, I., Van den Bergh, M.: Grothendieck groups and tilting objects. Algebra Represent. Theory 4, 1–23 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  60. Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. 39, 436–456 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  61. Ringel, C.M.: Tame algebras and integral quadratic forms. Springer Lecture Notes in Mathematics 1099 (1984)

  62. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Lecture Notes in Mathematics, vol. 269

  63. Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras 3: Representation-Infinite Tilted Algebras. London Math. Soc. Student Text., vol. 72. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  64. Stenström, B.: Rings of Quotients. Grundlehren der math. Wissensch, vol. 217. Springer, Berlin (1975)

    Google Scholar 

  65. Stovicek, J.: All \(n\)-cotilting modules are pure-injective. Proc. Am. Math. Soc. 134(7), 1891–1897 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. Tabuada, G.: Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories. C. R. Math. Acad. Sci. Paris 340(1), 15–19 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  67. Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque, Société Mathématique de France 239 (1996)

  68. Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics 38. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  69. Yang, D.: Recollements from generalized tilting. Proc. AMS 140(1), 83–91 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Facultad de Educación, Universidad de Murcia, Campus de Espinardo, 30100 , Murcia, Spain

    Pedro Nicolás

  2. Departamento de Matemáticas, Universidad de Murcia, Aptdo. 4021, Espinardo, 30100 , Murcia, Spain

    Manuel Saorín

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  1. Pedro Nicolás
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  2. Manuel Saorín
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Correspondence to Manuel Saorín.

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Dedicated to the memory of Michael Butler and Dieter Happel.

The authors are supported by research projects from the Ministerio de Economía y Competitividad of Spain (MTM2013-46837-P and MTM2016-77445-P) and from the Fundación ’Séneca’ of Murcia (19880/GERM/15), both with a part of FEDER funds.

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Nicolás, P., Saorín, M. Generalized Tilting Theory. Appl Categor Struct 26, 309–368 (2018). https://doi.org/10.1007/s10485-017-9495-x

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