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Selecta Mathematica

, Volume 24, Issue 5, pp 4555–4576 | Cite as

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

  • Benjamin Antieau
  • Akhil Mathew
  • Thomas Nikolaus
Article
  • 22 Downloads

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 

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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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blumberg, A.J., Gepner, D., Tabuada, G.: Uniqueness of the multiplicative cyclotomic trace. Adv. Math. 260, 191–232 (2014)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Goto, S., Yamagishi, K.: Finite generation of Noetherian graded rings. Proc. Am. Math. Soc. 89(1), 41–44 (1983)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hesselholt, L.: On the \(p\)-typical curves in Quillen’s \(K\)-theory. Acta Math. 177(1), 1–53 (1996)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  16. 16.
    Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 107(3), 447–452 (1992)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mathew, A., Naumann, N., Noel, J.: Nilpotence and descent in equivariant stable homotopy theory. Adv. Math. 305, 994–1084 (2017)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Scholze, P.: Canonical \(q\)-deformations in arithmetic geometry. Ann. Fac. Sci. Toulouse Math. 26(5), 1163–1192 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Schwede, S., Shipley, B.: Stable model categories are categories of modules. Topology 42(1), 103–153 (2003)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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

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