Mathematische Annalen

, Volume 353, Issue 3, pp 765–781 | Cite as

Approximations and adjoints in homotopy categories



We provide a criterion for the existence of right approximations in cocomplete additive categories; it is a straightforward generalisation of a result due to El Bashir. This criterion is used to construct adjoint functors in homotopy categories. Applications include the study of (pure) derived categories. For instance, it is shown that the pure derived category of any module category is compactly generated.


Exact Sequence Full Subcategory Module Category Compact Object Additive Category 
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  1. 1.
    Adámek J., Rosický J.: Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge (1994)CrossRefMATHGoogle Scholar
  2. 2.
    Adámek J., Rosický J.: On pure quotients and pure subobjects. Czechoslo. Math. J. 54(129)(3), 623–636 (2004)Google Scholar
  3. 3.
    Alonso Tarrío L., Jeremías López A., Souto Salorio M.J.: Localization in categories of complexes and unbounded resolutions. Can. J. Math. 52(2), 225–247 (2000)CrossRefMATHGoogle Scholar
  4. 4.
    Auslander M., Smalø S.O.: Preprojective modules over Artin algebras. J. Algebra 66(1), 61–122 (1980)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Beke T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129(3), 447–475 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Benson D.J.: Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras. Cambridge Studies in Advanced Mathematics, vol. 30. Cambridge University Press, Cambridge (1991)Google Scholar
  7. 7.
    Bican L., El Bashir R., Enochs E.: All modules have flat covers. Bull. Lond. Math. Soc. 33(4), 385–390 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bravo D., Enochs E.E., Iacob A.C., Jenda O.M.G., Rada J.: Cotorsion Pairs in C(R-Mod). PreprintGoogle Scholar
  9. 9.
    Casacuberta C., Neeman A.: Brown representability does not come for free. Math. Res. Lett. 16(1), 1–5 (2009)MathSciNetMATHGoogle Scholar
  10. 10.
    Christensen J.D., Hovey M.: Quillen model structures for relative homological algebra. Math. Proc. Camb. Philos. Soc. 133(2), 261–293 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Crawley-Boevey W.: Locally finitely presented additive categories. Commun. Algebra 22(5), 1641–1674 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    El Bashir R.: Covers and directed colimits. Algebras Represent. Theory 9(5), 423–430 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Enochs E.E.: Injective and flat covers, envelopes and resolvents. Israel J. Math. 39(3), 189–209 (1981)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Enochs E., Estrada S.: Relative homological algebra in the category of quasi-coherent sheaves. Adv. Math. 194(2), 284–295 (2005)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Franke J.: On the Brown representability theorem for triangulated categories. Topology 40(4), 667–680 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gabriel P., Ulmer F.: Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, vol. 221, Springer, Berlin (1971)Google Scholar
  17. 17.
    Gabriel P., Zisman M.: Calculus of Fractions and Homotopy Theory. Springer, New York (1967)MATHGoogle Scholar
  18. 18.
    Gruson L.: Simple coherent functors. In: Representation of Algebras (Ottowa, 1974). Lecture Notes in Mathematics, vol. 488, pp. 156–159. Springer, Berlin (1975)Google Scholar
  19. 19.
    Iyengar S., Krause H.: Acyclicity versus total acyclicity for complexes over Noetherian rings. Doc. Math. 11, 207–240 (2006)MathSciNetMATHGoogle Scholar
  20. 20.
    Jørgensen P.: The homotopy category of complexes of projective modules. Adv. Math. 193(1), 223–232 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Keller B.: Deriving DG categories. Ann. Sci. École Norm. Sup (4) 27(1), 63–102 (1994)MATHGoogle Scholar
  22. 22.
    Keller B.: Derived categories and their uses. In: Handbook of Algebra, vol. 1, pp. 671–701. North-Holland, Amsterdam (1996)Google Scholar
  23. 23.
    Kiełpiński R.: On Γ-pure injective modules. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15, 127–131 (1967)MATHGoogle Scholar
  24. 24.
    Krause H.: Exactly definable categories. J. Algebra 201(2), 456–492 (1998)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Krause H.: The stable derived category of a Noetherian scheme. Compos. Math. 141(5), 1128–1162 (2005)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ladkani S.: On derived equivalences of lines, rectangles and triangles. arXiv:0911.5137Google Scholar
  27. 27.
    Makkai M., Paré R.: Accessible Categories: The Foundations of Categorical Model Theory. Contemp. Math., vol. 104, Amer. Math. Soc., Providence (1989)Google Scholar
  28. 28.
    Maltsiniotis G.: Private communication. Luminy, Marseille Cedex (2010)Google Scholar
  29. 29.
    Murfet D.: The Mock Homotopy Category of Projectives and Grothendieck Duality. Ph.D. thesis, Canberra (2007)Google Scholar
  30. 30.
    Neeman A.: The derived category of an exact category. J. Algebra 135(2), 388–394 (1990)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Neeman A.: Triangulated Categories. Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton (2001)Google Scholar
  32. 32.
    Neeman A.: The homotopy category of flat modules, and Grothendieck duality. Invent. Math. 174(2), 255–308 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Neeman A.: Some adjoints in homotopy categories. Ann.Math (2) 171(3), 2143–2155 (2010)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Prest M.: Purity, spectra and localisation. Cambridge University Press, Cambridge (2009)MATHGoogle Scholar
  35. 35.
    Quillen D.: Higher algebraic K-theory. I. In: Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
  36. 36.
    Rada J., Saorin M.: Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra 26(3), 899–912 (1998)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Rickard J.: Morita theory for derived categories. J. Lond. Math. Soc. (2) 39(3), 436–456 (1989)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Rouquier R.: Complexes de chaînes étales et courbes de Deligne-Lusztig. J. Algebra 257(2), 482–508 (2002)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Schmeding A.: A construction of relatively pure submodules, in preparationGoogle Scholar

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Fakultät für Mathematik Universität BielefeldBielefeldGermany

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