Auslander–Buchweitz Approximation Theory for Triangulated Categories

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• 作者：Octavio Mendoza Hernández (1)
  Edith Corina Sáenz Valadez (2)
  Valente Santiago Vargas (1)
  María José Souto Salorio (3)
  
• 关键词：Triangulated categories ; Homological dimensions ; Approximation theory ; Primary 18E30 ; 18G20 ; Secondary 18G25
• 刊名：Applied Categorical Structures
• 出版年：2013
• 出版时间：April 2013
• 年：2013
• 卷：21
• 期：2
• 页码：119-139
• 全文大小：423 KB
• 参考文献：1. Auslander, M., Buchweitz, R.O.: The Homological theory of maximal Cohen–Macaulay approximations. Mem. Soc. Math. France 38, 5-7 (1989)
  2. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86, 111-52 (1991) external" href="http://dx.doi.org/10.1016/0001-8708(91)90037-8">CrossRef
  3. Auslander, M., Smalo, S.O.: Preprojective modules over artin algebra. J. Algebra 66, 61-22 (1980) external" href="http://dx.doi.org/10.1016/0021-8693(80)90113-1">CrossRef
  4. Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Asterisque 100, 5-71 (1982)
  5. Bondarko, M.V.: Weight structures for triangulated categories: weight filtrations, weight spectral sequences and weight complexes, applications to motives and to the stable homotopy category. Preprint available at ef="http://arxiv.org/abs/0704.4003v1">http://arxiv.org/abs/0704.4003v1
  6. Buan, A.B.: Subcategories of the derived category and cotilting complexes. Colloq. Math. 88(1), 1-1 (2001) external" href="http://dx.doi.org/10.4064/cm88-1-1">CrossRef
  7. Beligiannis, A., Reiten., I.: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 188, 883 (2007)
  8. Enochs, E.: Injective and flat covers, envelopes and resolvents. Isr. J. Math. 39, 189-09 (1981) external" href="http://dx.doi.org/10.1007/BF02760849">CrossRef
  9. Hashimoto, M.: Auslander-Buchweitz approximations of equivariant modules. LMS, Lecture Notes Series. vol. 282. Cambridge University Press (2000)
  10. Hügel, L.A., Mendoza, O.: Homological dimensions in cotorsion pairs. Ill. J. Math. 53(1), 251-63 (2009)
  11. Iyama, O., Yoshino., Y.: Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math 172, 117-68 (2008) external" href="http://dx.doi.org/10.1007/s00222-007-0096-4">CrossRef
  12. Krause, H.: Derived categories, resolutions, and Brown representability. Notes of a Course at the Summer School Interactions between Homotopy Theory and Algebra. Chicago (2004)
  13. Keller, B., Reiten, I.: Cluster tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123-51 (2007) external" href="http://dx.doi.org/10.1016/j.aim.2006.07.013">CrossRef
  14. Mendoza, O. Sáenz, C.: Tilting categories with applications to stratifying systems. J. Algebra 302, 419-49 (2006) external" href="http://dx.doi.org/10.1016/j.jalgebra.2006.01.006">CrossRef
  15. Mendoza, O., Sáenz, C., Santiago, V., Salorio, M.J.S.: Auslander–Buchwetiz context and co- / t-structures. Preprint, ef="http://arXiv.org/abs/math/1002.4604">arXiv:1002.4604 (2010)
  16. Pauksztello, D.: Compact corigid objects in triangulated categories and co-t-structures. Cent. Eur. J. Math. 6(1), 25-2 (2008) external" href="http://dx.doi.org/10.2478/s11533-008-0003-2">CrossRef
  17. Rouquier, R.: Dimension of triangulated categories. K-Theory 1, 193-56 (2008) external" href="http://dx.doi.org/10.1111/j.1755-2567.1935.tb00163.x">CrossRef
  
• 作者单位：Octavio Mendoza Hernández (1)
  Edith Corina Sáenz Valadez (2)
  Valente Santiago Vargas (1)
  María José Souto Salorio (3)
  
  1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito Exterior, C.P. 04510, México, D.F., Mexico
  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito Exterior, C.P. 04510, México, D.F., Mexico
  3. Facultade de Informática, Universidade da Coru?a, 15071, A Coru?a, Espa?a
  
• ISSN：1572-9095

文摘

We introduce and develop an analogous of the Auslander–Buchweitz approximation theory (see Auslander and Buchweitz, Societe Mathematique de France 38:5-7, 1989) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results concerning relative homological algebra in a triangulated category $\mathcal{T},$ which are based on the behavior of certain subcategories under finiteness of resolutions and vanishing of Hom-spaces. For example: we establish the existence of preenvelopes (and precovers) in certain triangulated subcategories of $\mathcal{T}.$ The results resemble various constructions and results of Auslander and Buchweitz, and are concentrated in exploring the structure of a triangulated category $\mathcal{T}$ equipped with a pair $(\mathcal{X},\omega),$ where $\mathcal{X}$ is closed under extensions and ω is a weak-cogenerator in $\mathcal{X},$ usually under additional conditions. This reduces, among other things, to the existence of distinguished triangles enjoying special properties, and the behavior of (suitably defined) (co)resolutions, projective or injective dimension of objects of $\mathcal{T}$ and the formation of orthogonal subcategories. Finally, some relationships with the Rouquier’s dimension in triangulated categories is discussed.