Relative homological dimensions in recollements of triangulated categories

doi:10.1007/s11464-019-0751-2

Front. Math. China

2019, Vol. 14

Issue (1) : 25-43 https://doi.org/10.1007/s11464-019-0751-2

RESEARCH ARTICLE

Relative homological dimensions in recollements of triangulated categories

Yonggang HU, Hailou YAO(

)

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China

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Abstract

Let $E$ be a proper class of triangles in a triangulated category $C$ , and let ( $A, B, C$ ) be a recollement of triangulated categories. Based on Beligiannis's work, we prove that $A$ and $C$ have enough $E$ -projective objects whenever $B$ does. Moreover, in this paper, we give the bounds for the $E$ -global dimension of $B$ in a recollement ( $A, B, C$ ) by controlling the behavior of the $E$ -global dimensions of the triangulated categories $A$ and $C$ : In particular, we show that the niteness of the $E$ -global dimensions of triangulated categories is invariant with respect to the recollements of triangulated categories.

Keywords Triangulated category proper class of triangles recollement E - global dimension')" href="#">

E

- global dimension derived category

Corresponding Author(s): Hailou YAO

Issue Date: 22 March 2019

Cite this article:

Yonggang HU,Hailou YAO. Relative homological dimensions in recollements of triangulated categories[J]. Front. Math. China, 2019, 14(1): 25-43.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0751-2
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I1/25

1	LAngeleri Hügel, S Koenig, Q HLiu. Recollements and tilting objects. J Pure Appl Algebra, 2011, 215: 420–438
2	LAngeleri Hügel, S Koenig, Q HLiu, DYang. Ladders and simplicity of derived module categories. J Algebra, 2017, 472: 15–66
3	JAsadollahi, S Salaria. Gorenstein objects in triangulated categories. J Algebra, 2004, 281: 264–286
4	JAsadollahi, S Salaria. Tate cohomology and Gorensteinness for triangulated categories. J Algebra, 2006, 299: 480–502
5	MAuslander, R O Buchweitz. The homological theory of maximal Cohen–Macaulay approximations. Mém Soc Math Fr (N S), 1989, 38: 5–37
6	A ABeilinson, J Bernstein, PDeligne. Faisceaux pervers, analysis and topology on singular spaces, I. Astérisque, 1982, 100: 5–171
7	ABeligiannis. Relative homological algebra and purity in triangulated categories. J Algebra, 2000, 227: 268–361
8	H XChen, C CXi. Recollements of derived categories II: algebraic K-theory. 2012, arXiv: 1203.5168
9	H XChen, C CXi. Recollements of derived categories III: nitistic dimensions. J Lond Math Soc (2), 2017, 95: 633–658
10	NGao. Gorensteinness, homological invariants and Gorenstein derived categories. Sci China Math, 2017, 60(3): 431–438
11	YHan. Recollements and Hochschild theory. J Algebra, 2014, 397: 535–547
12	DHappel. Reduction techniques for homological conjectures. Tsukuba J Math, 1993, 17: 115–130
13	BKeller. DerivingDG categories. Ann Sci Éc Norm Supér (4), 1994, 1: 63–102
14	SKoenig. Tilting complexes, perpendicular categories and recollements of derived module categories of rings. J Pure Appl Algebra, 1991, 73: 211–232
15	L PLi. Derived equivalences between triangular matrix algebras. Comm Algebra, 2018, 46(2): 615–628
16	RMacPherson, K Vilonen. Elementary construction of perverse sheaves. Invent Math, 1986, 84: 403–485
17	H RMargolis. Spectra and the Steenrod Algebra. North-Holland Math Library, Vol 29. Amsterdam: North-Holland Publishing Co, 1983
18	ANeeman. Triangulated Categories. Ann of Math Stud, Vol 148. Princeton: Princeton Univ Press, 2001
19	PNicolás. On torsion torsionfree triple. 2007, arXiv: 0801.0507
20	CPsaroudakis. Homological theory of recollements of abelian categories. J Algebra, 2014, 398: 63–110
21	Y YQin. Recollements and homological dimensions. Comm Algebra, 2018, 46(1): 1–12
22	Y YQin, YHan. Reducing homological conjectures by n-recollements. Algebr Represent Theory, 2016, 19: 377–395
23	WRen, Z KLiu. Gorenstein homological dimensions for triangulated categories. J Algebra, 2014, 410: 258–276
24	JRickard. Morita theory for derived categories. Bull Lond Math Soc, 1984, 16(5): 518–522
25	J LVerdier. Catégories dérivées, état 0. In: Lecture Notes in Math, Vol 569. Berlin: Springer-Verlag, 1977, 262–311
26	PZhang. Categorical resolutions of a class of derived categories. Sci China Math, 2018, 61: 1–12

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