Skip to main content
SpringerLink
Log in

A Simultaneous Generalization of Mutation and Recollement of Cotorsion Pairs on a Triangulated Category

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

In this article, we introduce the notion of concentric twin cotorsion pair on a triangulated category. This notion contains the notions of t-structure, cluster tilting subcategory, co-t-structure and functorally finite rigid subcategory as examples. Moreover, a recollement of triangulated categories can be regarded as a special case of concentric twin cotorsion pair. To any concentric twin cotorsion pair, we associate a pretriangulated subquotient category. This enables us to give a simultaneous generalization of the Iyama–Yoshino reduction and the recollement of cotorsion pairs. This allows us to give a generalized mutation on cotorsion pairs defined by the concentric twin cotorsion pair.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aihara, T., Iyama, O.: Silting mutation in triangulated categories. J. Lond. Math. Soc. 85(3), 633–668 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abe, N., Nakaoka, H.: General heart construction on a triangulated category (II): associated homological functor. Appl. Categ. Struct. 20(2), 161–174 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers (French) [Perverse sheaves] Analysis and topology on singular spaces, I (Luminy, 1981). Astérisque 100, 5–171 (1982)

    Google Scholar 

  4. Beligiannis, A., Reiten, I.: Homological and homotopical aspects of torsion theories. Mem. Am. Math. Soc. 188(883), viii+207 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Chen, J.: Cotorsion pairs in a recollement of triangulated categories. Commun. Algebra 41(8), 2903–2915 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)

    Book  MATH  Google Scholar 

  7. Hovey, M.: Cotorsion pairs, model category structures, and representation theory. Math. Z. 241(3), 553–592 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hovey, M.: Cotorsion pairs and model categories. Interactions between homotopy theory and algebra. Contemp. Math. 436, 277–296 (2007)

    Article  MATH  Google Scholar 

  9. Iyama, O., Kato, K., Miyachi, J.: Recollement of homotopy categories and Cohen–Macaulay modules. J. K-Theory 8(3), 507–542 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Iyama, O., Yang, D.: Silting reduction and Calabi–Yau reduction of triangulated categories. arXiv:1408.2678

  11. Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math. 172(1), 117–168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jørgensen, P.: Reflecting recollements. Osaka J. Math. 47(1), 209–213 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Keller, B., Reiten, I.: Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1), 123–151 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Koenig, S., Zhu, B.: From triangulated categories to abelian categories: cluster tilting in a general framework. Math. Z. 258(1), 143–160 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, Q., Vitória, J.: \(t\)-structures via recollements for piecewise hereditary algebras. J. Pure Appl. Algebra 216(4), 837–849 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nakaoka, H.: General heart construction on a triangulated category (I): unifying \(t\)-structures and cluster tilting subcategories. Appl. Categ. Struct. 19(6), 879–899 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nakaoka, H.: General heart construction for twin torsion pairs on triangulated categories. J. Algebra 374, 195–215 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nakaoka, H.: Equivalence of hearts of twin cotorsion pairs on triangulated categories. Commun. Algebra 44(10), 4302–4326 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nakaoka, H., Palu, Y.: Mutation via hovey twin cotorsion pairs and model structures in extriangulated categories. arXiv:1605.05607

  20. Pauksztello, D.: Compact corigid objects in triangulated categories and co-\(t\)-structures. Cent. Eur. J. Math. 6(1), 25–42 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhou, Y., Zhu, B.: Mutation of torsion pairs in triangulated categories and its geometric realization. arXiv:1105.3521

  22. Zhou, Y., Zhu, B.: \(T\)-structures and torsion pairs in a 2-Calabi–Yau triangulated category. J. Lond. Math. Soc. 89(1), 213–234 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research and Education Assembly, Science and Engineering Area, Research Field in Science, Kagoshima University, 1-21-35 Korimoto, Kagoshima, 890-0065, Japan

    Hiroyuki Nakaoka

Authors
  1. Hiroyuki Nakaoka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroyuki Nakaoka.

Additional information

Communicated by Martin Markl.

The author wishes to thank Professor Yann Palu for stimulating arguments, plenty of ideas, and useful comments and advices. The author wishes to thank Professor Steffen Koenig and Professor Jorge Vitória for introducing him the notion of recollement, and for their advices. The author wishes to thank Professor Osamu Iyama for introducing their results to him. This work is supported by JSPS KAKENHI Grant Numbers 25800022,  24540085.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nakaoka, H. A Simultaneous Generalization of Mutation and Recollement of Cotorsion Pairs on a Triangulated Category. Appl Categor Struct 26, 491–544 (2018). https://doi.org/10.1007/s10485-017-9501-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-017-9501-3

Keywords

Search

Navigation