Advertisement

Mathematische Zeitschrift

, Volume 283, Issue 3–4, pp 703–759 | Cite as

n-Abelian and n-exact categories

  • Gustavo JassoEmail author
Article

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.

Keywords

Abelian category Exact category Triangulated category n-Angulated category Homological algebra Cluster-tilting 

Mathematics Subject Classification

Primary 18E99 Secondary 18E10 18E30 

References

  1. 1.
    Amiot, C., Iyama, O., Reiten, I.: Stable categories of Cohen–Macaulay modules and cluster categories. arXiv:1104.3658 (2011)
  2. 2.
    Artin, M., Zhang, J.: Noncommutative projective schemes. Adv. Math. 109(2), 228–287 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auslander, M.: Coherent functors. In: Proceedings of Conference on Categorical Algebra (La Jolla, CA, 1965), pp. 189–231. Springer, New York (1966)Google Scholar
  4. 4.
    Auslander, M.: Representation Dimension of Artin Algebras. Lecture Notes. Queen Mary College, London (1971)Google Scholar
  5. 5.
    Auslander, M., Reiten, I.: Stable equivalence of dualizing r-varieties. Adv. Math. 12(3), 306–366 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Auslander, M., Reiten, I.: Applications of contravariantly finite subcategories. Adv. Math. 86(1), 111–152 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Auslander, M., Smalø, S.O.: Almost split sequences in subcategories. J. Algebra 69(2), 426–454 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)Google Scholar
  9. 9.
    Barot, M., Kussin, D., Lenzing, H.: The cluster category of a canonical algebra. Trans. Am. Math. Soc. 362(08), 4313–4330 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Beilinson, A.: Coherent sheaves on \(\text{ P }^d\) and problems in linear algebra. Funktsional. Anal. i Prilozhen 12(3), 68–69 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)Google Scholar
  12. 12.
    Bergh, P.A., Thaule, M.: The axioms for \(n\)-angulated categories. Algebr. Geom. Topol. 13(4), 2405–2428 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bergh, P.A., Thaule, M.: The Grothendieck group of an \(n\)-angulated category. J. Pure Appl. Algebra 218(2), 354–366 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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. 15.
    Bondal, A.I., Kapranov, M.M.: Framed triangulated categories. Math. Sb. 181(5), 669–683 (1990)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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. 18.
    Bühler, T.: Exact categories. Expo. Math. 28(1), 1–69 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fomin, S., Zelevinsky, A.: Cluster algebras i: foundations. J. Am. Math. Soc. 15(02), 497–529 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Frerick, L., Sieg, D.: Exact categories in functional analysis (2010). https://www.math.uni-trier.de/abteilung/analysis/HomAlg.pdf
  21. 21.
    Geiß, C., Keller, B., Oppermann, S.: \(n\)-angulated categories. J. Reine Angew. Math. 675, 101–120 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Geiß, C., Leclerc, B., Schröer, J.: Auslander algebras and initial seeds for cluster algebras. J. Lond. Math. Soc. 75(3), 718–740 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)Google Scholar
  24. 24.
    Grothendieck, A.: Sur quelques points d’algèbre homologique, i. Tohoku Math. J. (2) 9(2), 119–221 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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)Google Scholar
  26. 26.
    Herschend, M., Iyama, O.: Selfinjective quivers with potential and 2-representation-finite algebras. Compos. Math. 147(06), 1885–1920 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Herschend, M., Iyama, O., Minamoto, H., Oppermann, S.: Representation theory of Geigle–Lenzing complete intersections. arXiv:1409.0668 (2014)
  28. 28.
    Herschend, M., Iyama, O., Oppermann, S.: \(n\)-representation infinite algebras. Adv. Math. 252, 292–342 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Iyama, O.: Auslander correspondence. Adv. Math. 210(1), 51–82 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Iyama, O.: Higher-dimensional Auslander–Reiten theory on maximal orthogonal subcategories. Adv. Math. 210(1), 22–50 (2007b)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Iyama, O.: Cluster tilting for higher auslander algebras. Adv. Math. 226(1), 1–61 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Iyama, O., Oppermann, S.: \(n\)-Representation-finite algebras and \(n\)-APR tilting. Trans. Am. Math. Soc. 363(12), 6575–6614 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Iyama, O., Oppermann, S.: Stable categories of higher preprojective algebras. Adv. Math. 244, 23–68 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Iyama, O., Yoshino, Y.: Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math. 172(1), 117–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Jasso, G.: \(\tau ^2\)-Stable tilting complexes over weighted projective lines. arXiv:1402.6036 (2014)
  36. 36.
    Keller, B.: Chain complexes and stable categories. Manuscripta Math. 67(1), 379–417 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Keller, B.: On differential graded categories. In: International Congress of Mathematicians. Vol. II, pp. 151–190. Eur. Math. Soc., Zürich (2006)Google Scholar
  38. 38.
    Keller, B., Reiten, I.: Acyclic Calabi–Yau categories. Compos. Math. 144(05), 1332–1348 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Keller, B., Vossieck, D.: Sous les catégories dérivées. C. R. Acad. Sci. Paris Sér. I Math. 305(6), 22–228 (1987)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Minamoto, H.: Ampleness of two-sided tilting complexes. Int. Math. Res. Not. 2012(1), 67–101 (2012)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Neeman, A.: The derived category of an exact category. J. Algebra 135(2), 388–394 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 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)Google Scholar
  43. 43.
    Ringel, C.M.: The self-injective cluster-tilted algebras. Arch. Math. 91(3), 218–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 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)Google Scholar
  45. 45.
    Weibel, C.A.: An Introduction to Homological Algebra, Volume 38 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  46. 46.
    Yoshino, Y.: Cohen–Macaulay Modules over Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.Mathematik ZentrumUniversität BonnBonnGermany

Personalised recommendations