Skip to main content
SpringerLink
Log in

Descent in triangulated categories

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We establish effective descent for faithful ring objects in tensor triangulated categories. More generally, we discuss descent for monads in triangulated categories without tensor, where the answer is more subtle.

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. Balmer P., Schlichting M.: Idempotent completion of triangulated categories. J. Algebra 236(2), 819–834 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beck J.M.: Triples, algebras and cohomology. Repr. Theory Appl. Categ. 2, 1–59 (2003) (1967 thesis reprinted)

    Google Scholar 

  3. Borceux, F.: Handbook of categorical algebra. 2. In: Encyclopedia of Mathematics and its Applications, vol. 51. Cambridge University Press, Cambridge (1994)

  4. Cipolla M.: Discesa fedelmente piatta dei moduli. Rend. Circ. Mat. Palermo (2) 25(1–2), 43–46 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grothendieck, A.: Technique de descente et théorèmes d’existence en géometrie algébrique. I. Généralités. Descente par morphismes fidèlement plats. In: Séminaire Bourbaki, vol. 5, pages Exp. No. 190, 299–327. Soc. Math. France, Paris (1995)

  6. Hess, K.: A general framework for homotopic descent and codescent. Preprint. \({{\tt arXiv:1001.1556v2}}\) (2010)

  7. Kashiwara, M., Schapira, P.: Categories and sheaves. Grundlehren der Mathematischen Wissenschaften, vol. 332. Springer, Berlin (2006)

  8. Knus, M.-A., Ojanguren, M.: Théorie de la descente et algèbres d’Azumaya. Lecture Notes in Mathematics, vol. 389. Springer, Berlin (1974)

  9. Lurie, J.: Derived algebraic geometry II: Noncommutative algebra. Preprint, 175 pages. \({{\tt arXiv:math/0702299v5}}\) (2009)

  10. Mac Lane, S.: Categories for the working mathematician. In: Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)

  11. Mesablishvili B.: Pure morphisms of commutative rings are effective descent morphisms for modules—a new proof. Theory Appl. Categ. 7(3), 38–42 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Mesablishvili B.: Monads of effective descent type and comonadicity. Theory Appl. Categ. 16(1), 1–45 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Mesablishvili, B.: Notes on monads of effective descent type. (2010, in preparation)

  14. Neeman, A.: Triangulated categories. In: Annals of Mathematics Studies, vol. 148. Princeton University Press (2001)

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, UCLA, Los Angeles, CA, 90095-1555, USA

    Paul Balmer

Authors
  1. Paul Balmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Balmer.

Additional information

P. Balmer’s research was supported by NSF grant 0969644.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Balmer, P. Descent in triangulated categories. Math. Ann. 353, 109–125 (2012). https://doi.org/10.1007/s00208-011-0674-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-011-0674-z

Mathematics Subject Classification (2010)

Search

Navigation