Torsion pairs and recollements of extriangulated categories

In this article, we prove that if (A,B, C) is a recollement of extriangulated categories, then torsion pairs in A and C can induce torsion pairs in B, and the converse holds under natural assumptions. Besides, we give mild conditions on a cluster tilting subcategory on the middle category of a recollement of extriangulated categories, for the corresponding abelian quotients to form a recollement of abelian categories. 2020 Mathematics Subject Classification: 18G80; 18E10; 18E40.

[1]  Teimuraz Pirashvili,et al.  Comparison of Abelian Categories Recollements , 2004 .

[2]  Yann Palu,et al.  External triangulation of the homotopy category of exact quasi-category , 2020, 2004.02479.

[3]  Hiroyuki Nakaoka,et al.  General Heart Construction on a Triangulated Category (I): Unifying t-Structures and Cluster Tilting Subcategories , 2009, Appl. Categorical Struct..

[4]  Jiangsheng Hu,et al.  Proper classes and Gorensteinness in extriangulated categories , 2019, Journal of Algebra.

[5]  Jianmin Chen.  Cotorsion Pairs in a Recollement of Triangulated Categories , 2013 .

[6]  Xin Ma,et al.  Torsion pairs in recollements of abelian categories , 2017, Frontiers of Mathematics in China.

[7]  Chrysostomos Psaroudakis,et al.  Homological theory of recollements of abelian categories , 2014 .

[8]  R. Macpherson,et al.  Elementary construction of perverse sheaves , 1986 .

[9]  Panyue Zhou,et al.  Triangulated quotient categories revisited , 2016, 1608.00297.

[10]  Theo Buehler,et al.  Exact Categories , 2008, 0811.1480.

[11]  S. E. Dickson.  A torsion theory for Abelian categories , 1966 .

[12]  From triangulated categories to abelian categories: cluster tilting in a general framework , 2006, math/0605100.

[13]  Haicheng Zhang,et al.  Recollements of extriangulated categories , 2020, Colloquium Mathematicum.

[14]  Panyue Zhou,et al.  Abelian quotients of extriangulated categories , 2019, Proceedings - Mathematical Sciences.

[15]  Mutation in triangulated categories and rigid Cohen–Macaulay modules , 2006, math/0607736.

[16]  Wen Chang,et al.  Cluster Subalgebras and Cotorsion Pairs in Frobenius Extriangulated Categories , 2016, Algebras and Representation Theory.

[17]  Panyue Zhou,et al.  CLUSTER-TILTING SUBCATEGORIES IN EXTRIANGULATED CATEGORIES , 2019 .

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  Yann Palu,et al.  Extriangulated categories, Hovey twin cotorsion pairs and model structures , 2019 .

[20]  L. Salce.  Cotorsion theories for abelian groups , 1979 .

[21]  Raphael Bennett-Tennenhaus,et al.  Transport of structure in higher homological algebra , 2021 .

[22]  Yanan Lin,et al.  From recollement of triangulated categories to recollement of abelian categories , 2010 .