Relative homological dimensions in recollements of triangulated categories

Abstract

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

Key words： Triangulated category; proper class of triangles; recollement; \mathcal{E} - global dimension; derived category

References

1 Angeleri Hügel L, Koenig S, Liu Q H. Recollements and tilting objects. J Pure Appl Algebra, 2011, 215: 420–438

2 Angeleri Hügel L, Koenig S, Liu Q H, Yang D. Ladders and simplicity of derived module categories. J Algebra, 2017, 472: 15–66

3 Asadollahi J, Salaria S. Gorenstein objects in triangulated categories. J Algebra, 2004, 281: 264–286

4 Asadollahi J, Salaria S. Tate cohomology and Gorensteinness for triangulated categories. J Algebra, 2006, 299: 480–502

5 Auslander M, Buchweitz R O. The homological theory of maximal Cohen–Macaulay approximations. Mém Soc Math Fr (N S), 1989, 38: 5–37

6 Beilinson A A, Bernstein J, Deligne P. Faisceaux pervers, analysis and topology on singular spaces, I. Astérisque, 1982, 100: 5–171

7 Beligiannis A. Relative homological algebra and purity in triangulated categories. J Algebra, 2000, 227: 268–361

8 Chen H X, Xi C C. Recollements of derived categories II: algebraic K-theory. 2012, arXiv: 1203.5168

9 Chen H X, Xi C C. Recollements of derived categories III: nitistic dimensions. J Lond Math Soc (2), 2017, 95: 633–658

10 Gao N. Gorensteinness, homological invariants and Gorenstein derived categories. Sci China Math, 2017, 60(3): 431–438

11 Han Y. Recollements and Hochschild theory. J Algebra, 2014, 397: 535–547

12 Happel D. Reduction techniques for homological conjectures. Tsukuba J Math, 1993, 17: 115–130

13 Keller B. DerivingDG categories. Ann Sci Éc Norm Supér (4), 1994, 1: 63–102

14 Koenig S. Tilting complexes, perpendicular categories and recollements of derived module categories of rings. J Pure Appl Algebra, 1991, 73: 211–232

15 Li L P. Derived equivalences between triangular matrix algebras. Comm Algebra, 2018, 46(2): 615–628

16 MacPherson R, Vilonen K. Elementary construction of perverse sheaves. Invent Math, 1986, 84: 403–485

17 Margolis H R. Spectra and the Steenrod Algebra. North-Holland Math Library, Vol 29. Amsterdam: North-Holland Publishing Co, 1983

18 Neeman A. Triangulated Categories. Ann of Math Stud, Vol 148. Princeton: Princeton Univ Press, 2001

19 Nicolás P. On torsion torsionfree triple. 2007, arXiv: 0801.0507

20 Psaroudakis C. Homological theory of recollements of abelian categories. J Algebra, 2014, 398: 63–110

21 Qin Y Y. Recollements and homological dimensions. Comm Algebra, 2018, 46(1): 1–12

22 Qin Y Y, Han Y. Reducing homological conjectures by n-recollements. Algebr Represent Theory, 2016, 19: 377–395

23 Ren W, Liu Z K. Gorenstein homological dimensions for triangulated categories. J Algebra, 2014, 410: 258–276

24 Rickard J. Morita theory for derived categories. Bull Lond Math Soc, 1984, 16(5): 518–522

25 Verdier J L. Catégories dérivées, état 0. In: Lecture Notes in Math, Vol 569. Berlin: Springer-Verlag, 1977, 262–311

26 Zhang P. Categorical resolutions of a class of derived categories. Sci China Math, 2018, 61: 1–12