Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T06:52:40.686Z Has data issue: false hasContentIssue false
  • English
  • Français

GORENSTEIN SILTING COMPLEXES

Part of: Modules, bimodules and ideals Homological methods Representation theory of rings and algebras

Published online by Cambridge University Press:  28 January 2021

WEIQING CAO  and
JIAQUN WEI
WEIQING CAO
Affiliation:
Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing210023China e-mails: caoweiqing18@163.com; weijiaqun@njnu.edu.cn
JIAQUN WEI
Affiliation:
Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing210023China e-mails: caoweiqing18@163.com; weijiaqun@njnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce and study the notion of Gorenstein silting complexes, which is a generalization of Gorenstein tilting modules in Gorenstein-derived categories. We give the equivalent characterization of Gorenstein silting complexes. We give a sufficient condition for a partial Gorenstein silting complex to have a complement.

MSC classification

Primary: 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality
Secondary: 16E05: Syzygies, resolutions, complexes 16E35: Derived categories 16G10: Representations of Artinian rings
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 122 - 135
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aihara, T. and Iyama, O., Silting mutation in triangulated categories, J. Lond. Math. Soc. 85(3) (2012), 633–668.CrossRefGoogle Scholar
Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras, Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 1, 65 (Cambridge University Press, 2006).CrossRefGoogle Scholar
Auslander, M. and Reiten, I., Applications of contravariantly finite subcategories, Adv. Math. 86(1) (1991), 111–152.CrossRefGoogle Scholar
Auslander, M. and Smalø, S. O., Preprojective modules over Artin algebras, J. Algebra 66(1) (1980), 61–122.CrossRefGoogle Scholar
Auslander, M. and Solberg, Ø., Relative homology and representation theory I: Reative homology and homologically finite subcategories, Commun. Algebra 21(9) (1993), 2995–3031.CrossRefGoogle Scholar
Auslander, M. and Solberg, Ø., Relative homology and representation theory II: Relative cotilting theory, Commun. Algebra 21(9) (1993), 3033–3079.CrossRefGoogle Scholar
Auslander, M. and Solberg, Ø., Relative homology and representation theory III: Cotilting modules and Wedderburn correspondence, Commun. Algebra 21(9)(1993), 3081–3097.CrossRefGoogle Scholar
Bazzoni, S., A characterization of n-cotilting and n-tilting modules, J. Algebra 273 (2004), 359–372.CrossRefGoogle Scholar
Brenner, S. and Butler, M., Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors, In: Dlad, V., Gabriel, P. (eds) Representation Theory II. Lecture Notes in Mathematics, Springer, Berlin, Heidelberg. vol. 832 (1980), 103–169.Google Scholar
Colpi, R. and Trlifaj, J., Tilting modules and tilting torsion theories, J. Algebra 178 (1995), 614–634.CrossRefGoogle Scholar
Eilenberg, S. and Moore, J. C., Foundations of relative homological algebra, Memoirs of the American Mathematical Society, vol. 55 (AMS, Providence, RI, 1965).Google Scholar
Enochs, E. E. and Jenda, O. M. G., Relative homological algebras, de Gruyter Expositions in Mathematics, vol. 30 (Walter de Gruyter and Co., New York, 2000).Google Scholar
Gao, N. and Zhang, P., Gorenstein derived categories, J. Algebra 323(2010), 2041–2057.CrossRefGoogle Scholar
Hernández, O. M., Sáenz Valadez, E. C., Vargas, V. S. and Salorio, M. J. S., Auslander-Buchweitz context and co-t-structures, Appl. Categ. Struct. 21(5) (2013), 417–440.CrossRefGoogle Scholar
Hirotaka, K., On partial tilting complexes, Commun. Algebra 39(7) (2011), 2417–2429.Google Scholar
Keller, B. and Vossieck, D., Aisles in derived categories, Bull. Soc. Math. Belg. S¨¦r. A 40(2) (1988), 239–253.Google Scholar
Koenig, S. and Yang, D., Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438.Google Scholar
Miyashita, Y., Tilting modules of finite projective dimension, Math. Z. 193 (1986), 113–146.CrossRefGoogle Scholar
Moradifar, P. and Yassemi, S., Infinitely generated Gorenstein tilting modules, arXiv:1812.09349.Google Scholar
Rickard, M., Morita theory for derived categories, J. Lon. Math. Soc. 39(2) (1989), 436–456.CrossRefGoogle Scholar
Wei, J., Semi-tilting complexes, Israel J. Math. 194 (2013), 871–893.CrossRefGoogle Scholar
Yan, L., Li, W. and Ouyang, B., Gorenstein cotilting and tilting modules, Commun. Algebra 44(2) (2015), 591–603.CrossRefGoogle Scholar