Torsion pairs in recollements of abelian categories
Xin MA1, Zhaoyong HUANG2()
1. College of Science, Henan University of Engineering, Zhengzhou 451191, China 2. Department of Mathematics, Nanjing University, Nanjing 210093, China
For a recollement (A ;ℬ; C ) of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in ℬ; and the converse holds true under certain conditions.
Auslander M, Reiten I, Smalø S O. Representation Theory of Artin Algebras. Cambridge Stud Adv Math, Vol 36. Cambridge: Cambridge Univ Press, 1997
2
Beĭlinson A A, Bernstein J, Deligne P. Faisceaux pervers, analysis and topology on singular spaces, I. Astérisque, 1982, 100: 5{171
3
Beligiannis A, Reiten I. Homological and Homotopical Aspects of Torsion Theories. Mem Amer Math Soc, Vol 188, No 883. Providence: Amer Math Soc, 2007
4
Chen J M. Cotorsion pairs in a recollement of triangulated categories. Comm Algebra, 2013, 41: 2903–2915
5
Cline E,Parshall B, Scott L. Derived categories and Morita theory. J Algebra, 1986, 104: 397–409
6
Cline E, Parshall B, Scott L. Finite-dimensional algebras and highest weight categories. J Reine Angew Math, 1988, 391: 85–99
7
Dickson S E. A torsion theory for Abelian categories. Trans Amer Math Soc, 1966, 121: 223–235
8
Franjou V, Pirashvili T. Comparison of abelian categories recollements. Doc Math, 2004, 9: 41–56
9
Gentle R. T.T.F. theories in abelian categories. Comm Algebra, 1988, 16: 877–908
10
Happel D, Reiten I, Smalø S O. Tilting in Abelian Categories and Quasitilted Algebras. Mem Amer Math Soc, Vol 120, No 575. Providence: Amer Math Soc, 1996
11
Iyama O, Yoshino Y. Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent Math, 2008, 172: 117–168
12
Jans J P. Some aspects of torsion. Pacific J Math, 1965, 15: 1249–1259
13
Juteau D. Decomposition numbers for perverse sheaves. Ann Inst Fourier (Grenoble), 2009, 59: 1177–1229
14
Kuhn N J. Generic representations of the finite general linear groups and the Steenrod algebra II. K-Theory, 1994, 8: 395–428
15
Kuhn N J. A stratification of generic representation theory and generalized Schur algebras. K-Theory, 2002, 26: 15–49
16
Lin Y N, Wang M X.From recollement of triangulated categories to recollement of abelian categories. Sci China Math, 2010, 53: 1111–1116
17
Lin Z Q, Wang M X. Koenig's theorem for recollements of module categories. Acta Math Sinica (Chin Ser), 2011, 54: 461–466 (in Chinese)
18
Liu Q H, Vitória J, Yang D. Gluing silting objects. Nagoya Math J, 2014, 216: 117–151
19
Pirashvili T I. Polynomial functors. Trudy Tbiliss Mat Inst Razmadze Akad Nauk Gruzin SSR, 1988, 91: 55–66
20
Psaroudakis C. Homological theory of recollements of abelian categories. J Algebra, 2014, 398: 63–110
21
Psaroudakis C, Skartsæterhagen Ø, SolbergØ. Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements. Trans Amer Math Soc (Ser B), 2014, 1: 45–95
22
Psaroudakis C, Vitória J. Recollements of module categories. Appl Categ Structures, 2014, 22: 579–593