Skip to main content
SpringerLink
Log in

n-Abelian and n-exact categories

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.

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

Notes

  1. One may consider the more general case when \(\Sigma :\mathcal {F}\rightarrow \mathcal {F}\) is an autoequivalence. As mentioned in [21, 2.2 Rmks.], it can be shown that the assumption of \(\Sigma \) being invertible is but a mild sacrifice, cf. [39, Sect. 2].

References

  1. Amiot, C., Iyama, O., Reiten, I.: Stable categories of Cohen–Macaulay modules and cluster categories. arXiv:1104.3658 (2011)

  2. Artin, M., Zhang, J.: Noncommutative projective schemes. Adv. Math. 109(2), 228–287 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

  4. Auslander, M.: Representation Dimension of Artin Algebras. Lecture Notes. Queen Mary College, London (1971)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Auslander, M., Unger, L.: Isolated singularities and existence of almost split sequences. In: Dlab, V., Gabriel, P., Michler, G. (eds.) Representation Theory II Groups and Orders, Number 1178 in Lecture Notes in Mathematics, pp. 194–242. Springer, Berlin (1986)

  9. Barot, M., Kussin, D., Lenzing, H.: The cluster category of a canonical algebra. Trans. Am. Math. Soc. 362(08), 4313–4330 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Beilinson, A.: Coherent sheaves on \(\text{ P }^d\) and problems in linear algebra. Funktsional. Anal. i Prilozhen 12(3), 68–69 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), volume 100 of Astérisque, pp. 5–171. Soc. Math. France, Paris (1982)

  12. Bergh, P.A., Thaule, M.: The axioms for \(n\)-angulated categories. Algebr. Geom. Topol. 13(4), 2405–2428 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bergh, P.A., Thaule, M.: The Grothendieck group of an \(n\)-angulated category. J. Pure Appl. Algebra 218(2), 354–366 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bergh, P.A., Jasso, G., Thaule, M.: Higher \(n\)-angulations from local rings. To appear in J. Lond. Math. Soc. (2). arXiv:1311.2089 [math] (2013)

  15. Bondal, A.I., Kapranov, M.M.: Framed triangulated categories. Math. Sb. 181(5), 669–683 (1990)

    MathSciNet  MATH  Google Scholar 

  16. Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Buchweitz, R.-O.: Maximal Cohen–Macaulay Modules and Tate-Cohomology Over Gorenstein Rings. University of Hannover (1986). https://tspace.library.utoronto.ca/handle/1807/16682

  18. Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fomin, S., Zelevinsky, A.: Cluster algebras i: foundations. J. Am. Math. Soc. 15(02), 497–529 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Frerick, L., Sieg, D.: Exact categories in functional analysis (2010). https://www.math.uni-trier.de/abteilung/analysis/HomAlg.pdf

  21. Geiß, C., Keller, B., Oppermann, S.: \(n\)-angulated categories. J. Reine Angew. Math. 675, 101–120 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Geiß, C., Leclerc, B., Schröer, J.: Auslander algebras and initial seeds for cluster algebras. J. Lond. Math. Soc. 75(3), 718–740 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Geiß, C., Leclerc, B., Schröer, J.: Preprojective algebras and cluster algebras. In: Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep., pp. 253–283. Eur. Math. Soc., Zürich (2008)

  24. Grothendieck, A.: Sur quelques points d’algèbre homologique, i. Tohoku Math. J. (2) 9(2), 119–221 (1957)

    Article  MathSciNet  MATH  Google Scholar 

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

  26. Herschend, M., Iyama, O.: Selfinjective quivers with potential and 2-representation-finite algebras. Compos. Math. 147(06), 1885–1920 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Herschend, M., Iyama, O., Minamoto, H., Oppermann, S.: Representation theory of Geigle–Lenzing complete intersections. arXiv:1409.0668 (2014)

  28. Herschend, M., Iyama, O., Oppermann, S.: \(n\)-representation infinite algebras. Adv. Math. 252, 292–342 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Iyama, O., Oppermann, S.: \(n\)-Representation-finite algebras and \(n\)-APR tilting. Trans. Am. Math. Soc. 363(12), 6575–6614 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Iyama, O., Oppermann, S.: Stable categories of higher preprojective algebras. Adv. Math. 244, 23–68 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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 

  35. Jasso, G.: \(\tau ^2\)-Stable tilting complexes over weighted projective lines. arXiv:1402.6036 (2014)

  36. Keller, B.: Chain complexes and stable categories. Manuscripta Math. 67(1), 379–417 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Keller, B.: On differential graded categories. In: International Congress of Mathematicians. Vol. II, pp. 151–190. Eur. Math. Soc., Zürich (2006)

  38. Keller, B., Reiten, I.: Acyclic Calabi–Yau categories. Compos. Math. 144(05), 1332–1348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Keller, B., Vossieck, D.: Sous les catégories dérivées. C. R. Acad. Sci. Paris Sér. I Math. 305(6), 22–228 (1987)

    MathSciNet  MATH  Google Scholar 

  40. Minamoto, H.: Ampleness of two-sided tilting complexes. Int. Math. Res. Not. 2012(1), 67–101 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Neeman, A.: The derived category of an exact category. J. Algebra 135(2), 388–394 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  42. Quillen, D.: Higher algebraic \(k\)-theory. i. In Algebraic \(K\)-theory, I: Higher \(K\)-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), number 341 in Lecture Notes in Math., pp. 85–147. Springer, Berlin (1973)

  43. Ringel, C.M.: The self-injective cluster-tilted algebras. Arch. Math. 91(3), 218–225 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque, (239):xii + 253 pp. With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis (1996)

  45. Weibel, C.A.: An Introduction to Homological Algebra, Volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  46. Yoshino, Y.: Cohen–Macaulay Modules over Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

Download references

Author information

Author notes

  1. Gustavo Jasso

    Present address: Mathematik Zentrum, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

Authors and Affiliations

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

    Gustavo Jasso

Authors
  1. Gustavo Jasso
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gustavo Jasso.

Additional information

The author wishes to thank Erik Darpö, Laurent Demonet, Martin Herschend, Martin Kalck, Julian Külshammer, Boris Lerner, Yann Palu and Pierre-Guy Plamondon for motivating conversations regarding the contents of this article. This acknowledgment is extended to Prof. Osamu Iyama for his encouragement, helpful and interesting discussions, his comments on previous versions of this article, and especially for his generosity in sharing his ideas regarding a first definition of n-abelian category. Finally, the author wishes to express his sincere thanks to the anonymous referee for her/his detailed comments on a previous version of this article; in particular, for pointing out an error in earlier formulations of Theorem 3.20 and Lemma 3.22.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jasso, G. n-Abelian and n-exact categories. Math. Z. 283, 703–759 (2016). https://doi.org/10.1007/s00209-016-1619-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1619-8

Keywords

Mathematics Subject Classification

Search

Navigation