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Science China Mathematics

, Volume 59, Issue 5, pp 861–874 | Cite as

Applications of balanced pairs

  • HuanHuan Li
  • JunFu Wang
  • ZhaoYong Huang
Articles
  • 65 Downloads

Abstract

Let (X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to (X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y (resp. Y-coresolution dimension of X) is finite, then the bounded homotopy category of Y (resp. X) is contained in that of X (resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.

Keywords

balanced pairs relative cotorsion pairs relative derived categories relative singularity categories relative (co)resolution dimension 

MSC(2010)

16G25 18G10 18G20 

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References

  1. 1.
    Asadollahi J, Hafezi R, Vahed R. Gorenstein derived equivalences and their invariants. J Pure Appl Algebra, 2014, 218: 888–903MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auslander M, Reiten I. Applications of contravariantly finite subcategories. Adv Math, 1991, 86: 111–152MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Buchweitz R O. Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings. Unpublished manuscript, http://hdl.handle.net/1807/16682, 1986Google Scholar
  4. 4.
    Cartan H, Eilenberg S. Homological Algebra. Princeton: Princeton University Press, 1956zbMATHGoogle Scholar
  5. 5.
    Chen X W. Homotopy equivalences induced by balanced pairs. J Algebra, 2010, 324: 2718–2731MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen X W. Relative singularity categories and Gorenstein-projective modules. Math Nachr, 2011, 284: 199–212MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen X W, Zhang P. Quotient triangulated categories. Manuscripta Math, 2007, 123: 167–183MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Christensen LW, Frankild A, Holm H. On Gorenstein projective, injective and flat dimensions—a functorial description with applications. J Algebra, 2006, 302: 231–279MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Enochs E E, Jenda O M G. Balanced functors applied to modules. J Algebra, 1985, 92: 303–310MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Enochs E E, Jenda O M G. Relative Homological Algebra. Berlin-New York: Walter de Gruyter, 2000CrossRefzbMATHGoogle Scholar
  11. 11.
    Enochs E E, Jenda O M G, Torrecillas B, et al. Torsion theory relative to Ext. Http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.8694, 1998Google Scholar
  12. 12.
    Gao N, Zhang P. Gorenstein derived categories. J Algebra, 2010, 323: 2041–2057MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Happel D. On Gorenstein algebras. In: Representation Theory of Finite Groups and Finite-Dimensional Algebras. Progress in Mathematics, vol. 95. Basel: Birkhäuser, 1991, 389–404MathSciNetzbMATHGoogle Scholar
  14. 14.
    Hovey M. Cotorsion pairs and model categories. In: Interactions Between Homotopy Theory and Algebra. Contemp Mathematics, vol. 436. Providence, RI: Amer Math Soc, 2007, 277–296MathSciNetzbMATHGoogle Scholar
  15. 15.
    Huang Z Y, Iyama O. Auslander-type conditions and cotorsion pairs. J Algebra, 2007, 318: 93–110MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iyama O, Yoshino Y. Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent Math, 2008, 172: 117–168MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Krause H, Solberg Ø. Applications of cotorsion pairs. J Lond Math Soc, 2003, 68: 631–650MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Orlov D. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc Steklov Inst Math, 2004, 246: 227–248MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rickard J. Derived categories and stable equivalence. J Pure Appl Algebra, 1989, 61: 303–317MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Salce L. Cotorsion Theories for Abelian Categories. Cambridge: Cambridge University Press, 1979zbMATHGoogle Scholar
  21. 21.
    Verdier J L. Catégories dérivées. Etat 0. In: Lecture Notes in Math, vol. 569. Berlin: Springer-Verlag, 1977, 262–311MathSciNetGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina

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