Advertisement

Selecta Mathematica

, 25:13 | Cite as

Tensor-triangular fields: ruminations

  • Paul Balmer
  • Henning Krause
  • Greg StevensonEmail author
Open Access
Article

Abstract

We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable homological tensor-functors to Grothendieck categories.

Keywords

Residue field tt-Geometry Module category 

Mathematics Subject Classification

18E30 (20J05, 55U35) 

Notes

References

  1. 1.
    Auslander, M.: Representation theory of Artin algebras. II. Commun. Algebra 1, 269–310 (1974)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balmer, P.: Spectra, spectra, spectra–tensor triangular spectra versus Zariski spectra of endomorphism rings. Algebra Geom. Topol. 10(3), 1521–1563 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Balmer, P.: Tensor triangular geometry. In: International Congress of Mathematicians, Hyderabad (2010), vol. II, pp. 85–112. Hindustan Book Agency (2010)Google Scholar
  4. 4.
    Balmer, P.: Nilpotence theorems via homological residue fields. Preprint, 16 pp. arXiv:1710.04799 (2017)
  5. 5.
    Balmer, P., Dell’Ambrogio, I., Sanders, B.: Restriction to finite-index subgroups as étale extensions in topology KK-theory and geometry. Algebra Geom. Topol. 15(5), 3025–3047 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Balmer, P., Dell’Ambrogio, I., Sanders, B.: Grothendieck-Neeman duality and the Wirthmüller isomorphism. Compos. Math. 152(8), 1740–1776 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Beligiannis, A.: Relative homological algebra and purity in triangulated categories. J. Algebra 227(1), 268–361 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Balmer, P., Favi, G.: Generalized tensor idempotents and the telescope conjecture. Proc. Lond. Math. Soc. 102(6), 1161–1185 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Balmer, P., Krause, H., Stevenson, G.: The frame of smashing tensor-ideals. Math. Proc. Cambridge Philos. Soc. (2018).  https://doi.org/10.1017/S0305004118000725
  10. 10.
    Carlson, J.F., Friedlander, E.M., Pevtsova, J.: Modules of constant Jordan type. J. Reine Angew. Math. 614, 191–234 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. Fr. 90, 323–448 (1962)CrossRefGoogle Scholar
  12. 12.
    Grothendieck, A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J. 2(9), 119–221 (1957)zbMATHGoogle Scholar
  13. 13.
    Heller, J., Ormsby, K.M.: Primes and fields in stable motivic homotopy theory. Geom. Topol. 22(4), 2187–2218 (2018).  https://doi.org/10.2140/gt.2018.22.2187
  14. 14.
    Hovey, M., Palmieri, J.H.: The structure of the Bousfield lattice. In: Boardman, J.M., Meyer, J-P., Morava, J., Stephen Wilson, W. (eds.) Homotopy Invariant Algebraic Structures (Baltimore, MD, 1998), Contemporary Mathematics, vol. 239, pp. 175–196. American Mathematical Society, Providence, RI (1999)Google Scholar
  15. 15.
    Hovey, M., Palmieri, J.H., Strickland, N.P.: Axiomatic Stable Homotopy Theory. Memoirs of the American Mathematical Society, vol. 128(610) (1997)Google Scholar
  16. 16.
    Krause, H., Reichenbach, U.: Endofiniteness in stable homotopy theory. Trans. Am. Math. Soc. 353(1), 157–173 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Krause, Henning: Smashing subcategories and the telescope conjecture—an algebraic approach. Invent. Math. 139(1), 99–133 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mathew, A.: Residue fields for a class of rational \({\text{ E }}_\infty \)-rings andapplications. J. Pure Appl. Algebra 221(3), 707–748 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mac Lane, S.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)zbMATHGoogle Scholar
  20. 20.
    Neeman, A.: The chromatic tower for \(D(R)\). Topology 31(3), 519–532 (1992)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Amnon, N.: The connection between the \(K\)-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel. Ann. Sci. École Norm. Sup. 25(5), 547–566 (1992)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Neeman, A.: Triangulated Categories. Annals of Mathematics Studies, vol. 148. Princeton University Press, Princeton (2001)Google Scholar
  23. 23.
    Claus Michael Ringel and Hiroyuki Tachikawa: \({\text{ QF }}-3\) rings. J. Reine Angew. Math. 272, 49–72 (1974)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Stevenson, G.: A tour of support theory for triangulated categories through tensor triangular geometry. In: Building Bridges Between Algebra and Topology, pp. 63–101. Springer (2018)Google Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematics DepartmentUCLALos AngelesUSA
  2. 2.Universität Bielefeld, Fakultät für MathematikBielefeldGermany
  3. 3.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK

Personalised recommendations