Recollements arising from cotorsion pairs on extriangulated categories

Abstract

This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category. As an application, this result generalizes the work by W. J. Chen, Z. K. Liu, and X. Y. Yang in a triangulated case [J. Algebra Appl., 2018, 17(5): 1–15]. Moreover, it highlights new phenomena when it applied to an exact category. Finally, we give some applications of our main results. In particular, we obtain Krause's recollement whose proofs are both elementary and very general.

Keywords

Extriangulated categories / recollements / cotorsion pairs / adjoint pairs

References

[1] Beilinson A, Bernstein J, Deligne P. Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981). Paris: Soc Math France, 1982, 5–171

[2] Borceux F. Handbook of Categorical Algebra 1, Basic Category Theory. Encyclopedia Math Appl, Vol 50. Cambridge: Cambridge Univ Press, 1994

[3] Chen W J, Liu Z K, Yang X Y. Recollements associated to cotorsion pairs. J Algebra Appl, 2018, 17(5): 1–15

[4] Enochs E E, Jenda O M G. Relative Homological Algebra, Vol 2. Berlin: De Gruyter, 2011

[5] Gillespie J. The at model structure on Ch(R): Trans Amer Math Soc, 2004, 356(8): 3369–3390

[6] Gillespie J. Cotorsion pairs and degreewise homological model structures. Homology Homotopy Appl, 2008, 10(1): 283–304

[7] Gillespie J. Gorenstein complexes and recollements from cotorsion pairs. Adv Math, 2016, 291: 859–911

[8] Göbel R, Trlifaj J. Approximations and Endomorphism Algebras of Modules. Berlin: Walter de Gruyter, 2006

[9] Liu Y, Nakaoka H. Hearts of twin cotorsion pairs on extriangulated categories. J Algebra, 2019, 528: 96–149

[10] MacPherson R, Vilonen K. Elementary construction of perverse sheaves. Invent Math, 1986, 84(2): 403–435

[11] Nakaoka H, Palu Y. Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah Topol Géom Différ Catéeg, 2019, 60(2): 117–193

[12] Wang M X, Lin Z Q. Recollement of additive quotient categories. arXiv: 1502.00479

[13] Zheng Q L, Wei J Q. One-sided triangulated categories induced by concentric twin cotorsion pairs. J Algebra Appl, 2020, 19(8): 2050142

[14] Zhou P Y, Zhu B. Triangulated quotient categories revisited. J Algebra, 2018, 502: 196–232

[15] Zhu B, Zhuang X. Tilting subcategories in extriangulated categories. Front Math China, 2020, 15(1): 225–253

RIGHTS & PERMISSIONS

2021 Higher Education Press