Skip to main content
SpringerLink
Log in

Recollements for Dualizing k-Varieties and Auslander’s Formulas

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Given the pair of a dualizing k-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander’s formulas: the first one realizes a module category as a Serre quotient of a suitable functor category. The second one shows a close connection between Auslander–Bridger sequences and recollements. The third one gives a new proof of the higher defect formula which includes the higher Auslander–Reiten duality as a special case.

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. Auslander, M.: Coherent functors. In: Proceedings of the Conference on Categorical Algebra (La Jolla, Calif., 1965), pp. 189–231. Springer, New York (1966)

  2. Auslander, M., Bridger, M.: Stable module theory. Memoirs of the American Mathematical Society, No. 94. American Mathematical Society, Providence (1969)

  3. Auslander, M., Reiten, I.: Stable equivalence of dualizing R-varieties. Adv. Math. 12(3), 306–366 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  4. Auslander, M., Reiten, I., Smalø, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  5. Auslander, M., Simson, D., Skowrónski, A.: Elements of the Representation Theory of Associative Algebras. Volume 1 of London Mathematical Society Student Texts 65. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  6. Auslander, M., Smalo, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers (perverse sheaves). Analysis and topology on singular spaces, I, Luminy, 1981. Asterisque 100, 5–171 (1982). (in French)

    Google Scholar 

  8. Buchweitz, R.-O.: Finite representation type and periodic Hochschild (co-)homology. Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997). In: Contemporary Mathematics, vol. 229, pp. 81–109. American Mathematical Society, Providence (1998)

  9. Crawley-Boevey, W., Sauter, J.: On quiver Grassmannians and orbit closures for representation-finite algebras. Math. Z. 285(1–2), 367–395 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eiríksson, Ö.: From submodule categories to the stable Auslander algebra. J. Algebra 486, 98–118 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Franjou, V., Pirashvili, T.: Comparison of abelian categories recollements. Doc. Math. 9, 41–56 (2004). (electronic)

    MathSciNet  MATH  Google Scholar 

  12. Freyd, P.: Representations in abelian categories. In: Proceedings of the Conference on Categorical Algebra (La Jolla, Calif., 1965) pp. 95–120 Springer, New York (1966)

  13. Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. France 90, 323–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  15. Iyama, O.: Auslander correspondence. Adv. Math. 210(1), 51–82 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Iyama, O.: Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories. Adv. Math. 210(1), 22–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Iyama, O.: Cluster tilting for higher Auslander algebras. Adv. Math. 226(1), 1–61 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iyama, O., Jasso, G.: Higher Auslander correspondence for dualizing \(R\)-varieties. Algebraic Represent. Theory 20(2), 335–354 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jasso, G.: \(n\)-abelian and \(n\)-exact categories. Math. Z. 283(3–4), 703–759 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jasso, G., Kvamme, S.: An introduction to higher Auslander–Reiten theory. arXiv:1610.05458v1 (2016)

  21. Krause, H.: Smashing subcategories and the telescope conjecture: an algebraic approach. Invent. Math. 139(1), 99–133 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Krause, H.: A short proof for Auslander’s defect formula. Linear Algebra Appl. 365, 267–270 (2003). Special issue on linear algebra methods in representation theory

  23. Krause, H.: Highest weight categories and recollements. Ann. Inst. Fourier (Grenoble) 67(6), 2679–2701 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lenzing, H.: Auslander’s work on artin algebras. In: Algebras and modules, I (Trondheim, 1996), CMS Conference Proceedings, vol. 23, pp. 83–105. American Mathematical Society, Providence (1998)

  25. Popescu, N.: Abelian Categories with Applications to Rings and Modules. London Mathematical Society Monographs, No. 3. Academic Press, London (1973)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Psaroudakis, C., Vitória, J.: Recollements of module categories. Appl. Categ. Struct. 22(4), 579–593 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

First and foremost, I would like to express my gratitude to my supervisors Kiriko Kato and Osamu Iyama, who gave me so many helpful suggestions and discussions. I am also grateful to Takahide Adachi for his valuable comments and advice.

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

    Yasuaki Ogawa

Authors
  1. Yasuaki Ogawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasuaki Ogawa.

Additional information

Henning Krause.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ogawa, Y. Recollements for Dualizing k-Varieties and Auslander’s Formulas. Appl Categor Struct 27, 125–143 (2019). https://doi.org/10.1007/s10485-018-9546-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-018-9546-y

Keywords

Mathematics Subject Classification

Search

Navigation