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.
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Asadollahi J, Hafezi R, Vahed R. Gorenstein derived equivalences and their invariants. J Pure Appl Algebra, 2014, 218: 888–903
Auslander M, Reiten I. Applications of contravariantly finite subcategories. Adv Math, 1991, 86: 111–152
Buchweitz R O. Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings. Unpublished manuscript, http://hdl.handle.net/1807/16682, 1986
Cartan H, Eilenberg S. Homological Algebra. Princeton: Princeton University Press, 1956
Chen X W. Homotopy equivalences induced by balanced pairs. J Algebra, 2010, 324: 2718–2731
Chen X W. Relative singularity categories and Gorenstein-projective modules. Math Nachr, 2011, 284: 199–212
Chen X W, Zhang P. Quotient triangulated categories. Manuscripta Math, 2007, 123: 167–183
Christensen LW, Frankild A, Holm H. On Gorenstein projective, injective and flat dimensions—a functorial description with applications. J Algebra, 2006, 302: 231–279
Enochs E E, Jenda O M G. Balanced functors applied to modules. J Algebra, 1985, 92: 303–310
Enochs E E, Jenda O M G. Relative Homological Algebra. Berlin-New York: Walter de Gruyter, 2000
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, 1998
Gao N, Zhang P. Gorenstein derived categories. J Algebra, 2010, 323: 2041–2057
Happel D. On Gorenstein algebras. In: Representation Theory of Finite Groups and Finite-Dimensional Algebras. Progress in Mathematics, vol. 95. Basel: Birkhäuser, 1991, 389–404
Hovey M. Cotorsion pairs and model categories. In: Interactions Between Homotopy Theory and Algebra. Contemp Mathematics, vol. 436. Providence, RI: Amer Math Soc, 2007, 277–296
Huang Z Y, Iyama O. Auslander-type conditions and cotorsion pairs. J Algebra, 2007, 318: 93–110
Iyama O, Yoshino Y. Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent Math, 2008, 172: 117–168
Krause H, Solberg Ø. Applications of cotorsion pairs. J Lond Math Soc, 2003, 68: 631–650
Orlov D. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. Proc Steklov Inst Math, 2004, 246: 227–248
Rickard J. Derived categories and stable equivalence. J Pure Appl Algebra, 1989, 61: 303–317
Salce L. Cotorsion Theories for Abelian Categories. Cambridge: Cambridge University Press, 1979
Verdier J L. Catégories dérivées. Etat 0. In: Lecture Notes in Math, vol. 569. Berlin: Springer-Verlag, 1977, 262–311
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Li, H., Wang, J. & Huang, Z. Applications of balanced pairs. Sci. China Math. 59, 861–874 (2016). https://doi.org/10.1007/s11425-015-5094-1
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DOI: https://doi.org/10.1007/s11425-015-5094-1
Keywords
- balanced pairs
- relative cotorsion pairs
- relative derived categories
- relative singularity categories
- relative (co)resolution dimension