Advertisement

On the Blumberg–Mandell Künneth theorem for \(\mathrm {TP}\)

  • Benjamin Antieau
  • Akhil Mathew
  • Thomas Nikolaus
Article

Abstract

We give a new proof of the recent Künneth theorem for periodic topological cyclic homology of smooth and proper dg categories over perfect fields of characteristic \(p>0\) due to Blumberg and Mandell. Our result is slightly stronger and implies a finiteness theorem for topological cyclic homology of such categories.

Keywords

Künneth theorems The Tate construction Topological Hochschild homology Periodic topological cyclic homology 

Mathematics Subject Classification

14F30 16E40 19D55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ayala, D., Mazel-Gee, A., Rozenblyum, N.: The geometry of the cyclotomic trace. arXiv eprints, (2017) https://arxiv.org/abs/1710.06409
  2. 2.
    Blumberg, A.J., Gepner, D., Tabuada, G.: A universal characterization of higher algebraic \(K\)-theory. Geom. Topol. 17(2), 733–838 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blumberg, A.J., Gepner, D., Tabuada, G.: Uniqueness of the multiplicative cyclotomic trace. Adv. Math. 260, 191–232 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blumberg, A., Mandell, M.: The strong Künneth theorem for topological periodic cyclic homology. (2017) http://arxiv.org/abs/1706.06846
  5. 5.
    Bhatt, B., Matthew, M., Scholze, P.: Integral \(p\)-adic Hodge theory and topological Hochschild homology. (2018) https://arxiv.org/abs/1802.03261
  6. 6.
    Dwyer, W.G., Greenlees, J.P.C.: Complete modules and torsion modules. Am. J. Math. 124(1), 199–220 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Geisser, T., Hesselholt, L.: Topological cyclic homology of schemes. In: Algebraic \(K\)-theory (Seattle, WA, 1997), Proceedings of symposia in pure mathematics, vol. 67, pp. 41–87. American Mathematical Society, Providence (1999)Google Scholar
  8. 8.
    Gros, M., Suwa, N.: La conjecture de Gersten pour les faisceaux de Hodge–Witt logarithmique. Duke Math. J. 57(2), 615–628 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goto, S., Yamagishi, K.: Finite generation of Noetherian graded rings. Proc. Am. Math. Soc. 89(1), 41–44 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hesselholt, L.: On the \(p\)-typical curves in Quillen’s \(K\)-theory. Acta Math. 177(1), 1–53 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hesselholt, L.: Topological hochschild homology and the Hasse–Weil zeta function. In: Alpine algebraic and applied topology (Saas Almagell, Switzerland, 2016), Contemporary Mathematics—American Mathematical Society, Providence (2016)Google Scholar
  12. 12.
    Hesselholt, L., Madsen, I.: On the \(K\)-theory of finite algebras over Witt vectors of perfect fields. Topology 36(1), 29–101 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hopkins, M.J., Mahowald, M., Sadofsky, H.: Constructions of elements in Picard groups. In: Topology and representation theory (Evanston, IL, 1992), Contemporary Mathematics, vol. 158 , pp. 89–126. American Mathematical Society, Providence (1994)Google Scholar
  14. 14.
    Illusie, L.: Complexe de de Rham–Witt et cohomologie cristalline. Ann. Sci. École Norm. Sup. (4) 12(4), 501–661 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Illusie, L., Raynaud, M.: Les suites spectrales associées au complexe de de Rham–Witt. Inst. Hautes Études Sci. Publ. Math. 57, 73–212 (1983)CrossRefzbMATHGoogle Scholar
  16. 16.
    Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107(3), 447–452 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lurie, J.: Derived algebraic geometry XII: Proper morphisms, completions, and the Grothendieck existence theorem. 2011. http://www.math.harvard.edu/~lurie/papers/DAG-XII.pdf (2011)
  18. 18.
    Lurie, J.: Higher algebra. http://www.math.harvard.edu/~lurie/papers/HA.pdf (2017)
  19. 19.
    Matijevic, J.: Three local conditions on a graded ring. Trans. Am. Math. Soc. 205, 275–284 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mathew, A., Naumann, N., Noel, J.: Nilpotence and descent in equivariant stable homotopy theory. Adv. Math. 305, 994–1084 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nikolaus, T., Scholze, P.: On topological cyclic homology. (2017) http://arxiv.org/abs/1707.01799
  22. 22.
    Năstăsescu, C., van Oystaeyen, F.: Graded ring theory. North-Holland Publishing Co., Amsterdam (1982)zbMATHGoogle Scholar
  23. 23.
    Quillen, D.: On the cohomology and \(K\)-theory of the general linear groups over a finite field. Ann. Math. 2(96), 552–586 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Scholze, P.: Canonical \(q\)-deformations in arithmetic geometry. Ann. Fac. Sci. Toulouse Math. 26(5), 1163–1192 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schwede, S., Shipley, B.: Stable model categories are categories of modules. Topology 42(1), 103–153 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tabuada, G.: Noncommutative motives in positive characteristic and their applications. (2017) http://arxiv.org/abs/1707.04248
  27. 27.
    Toën, B.: Derived Azumaya algebras and generators for twisted derived categories. Invent. Math. 189(3), 581–652 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Benjamin Antieau
    • 1
  • Akhil Mathew
    • 2
  • Thomas Nikolaus
    • 3
  1. 1.University of Illinois at ChicagoChicagoUSA
  2. 2.University of ChicagoChicagoUSA
  3. 3.Universität MünsterMünsterGermany

Personalised recommendations