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A short proof of telescopic Tate vanishing


Authors: Dustin Clausen and Akhil Mathew
Journal: Proc. Amer. Math. Soc. 145 (2017), 5413-5417
MSC (2010): Primary 55P42, 55P47
DOI: https://doi.org/10.1090/proc/13648
Published electronically: June 16, 2017
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Abstract: We give a short proof of a theorem of Kuhn that Tate constructions for finite group actions vanish in telescopically localized stable homotopy theory. In particular, we observe that Kuhn's theorem is equivalent to the statement that the transfer $ BC_{p+} \to S^0$ admits a section after telescopic localization, which in turn follows from the Kahn-Priddy theorem.


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  • [Bal05] Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149-168. MR 2196732, https://doi.org/10.1515/crll.2005.2005.588.149
  • [Bou01] A. K. Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391-2426. MR 1814075, https://doi.org/10.1090/S0002-9947-00-02649-0
  • [GM95] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178. MR 1230773, https://doi.org/10.1090/memo/0543
  • [GS96] J. P. C. Greenlees and Hal Sadofsky, The Tate spectrum of $ v_n$-periodic complex oriented theories, Math. Z. 222 (1996), no. 3, 391-405. MR 1400199, https://doi.org/10.1007/PL00004264
  • [HL13] Michael Hopkins and Jacob Lurie, Ambidexterity in $ K(n)$-local stable homotopy theory, available at http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf.
  • [HS96] Mark Hovey and Hal Sadofsky, Tate cohomology lowers chromatic Bousfield classes, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3579-3585. MR 1343699, https://doi.org/10.1090/S0002-9939-96-03495-8
  • [HS98] Michael J. Hopkins and Jeffrey H. Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. (2) 148 (1998), no. 1, 1-49. MR 1652975, https://doi.org/10.2307/120991
  • [KP78] Daniel S. Kahn and Stewart B. Priddy, The transfer and stable homotopy theory, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 103-111. MR 0464230, https://doi.org/10.1017/S0305004100054335
  • [Kuh89] Nicholas J. Kuhn, Morava $ K$-theories and infinite loop spaces, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 243-257. MR 1000381, https://doi.org/10.1007/BFb0085232
  • [Kuh04] Nicholas J. Kuhn, Tate cohomology and periodic localization of polynomial functors, Invent. Math. 157 (2004), no. 2, 345-370. MR 2076926, https://doi.org/10.1007/s00222-003-0354-z
  • [Kuh08] Nicholas J. Kuhn, A guide to telescopic functors, Homology Homotopy Appl. 10 (2008), no. 3, 291-319. MR 2475626
  • [Mah82] Mark Mahowald, The image of $ J$ in the $ EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65-112. MR 662118, https://doi.org/10.2307/2007048
  • [Mil81] Haynes R. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space, J. Pure Appl. Algebra 20 (1981), no. 3, 287-312. MR 604321, https://doi.org/10.1016/0022-4049(81)90064-5
  • [MNN17] Akhil Mathew, Niko Naumann, and Justin Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017), 994-1084. MR 3570153, https://doi.org/10.1016/j.aim.2016.09.027
  • [MS88] Mark Mahowald and Paul Shick, Root invariants and periodicity in stable homotopy theory, Bull. London Math. Soc. 20 (1988), no. 3, 262-266. MR 931189, https://doi.org/10.1112/blms/20.3.262
  • [Seg74] Graeme Segal, Operations in stable homotopy theory, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), Cambridge Univ. Press, London, 1974, pp. 105-110. London Math Soc. Lecture Note Ser., No. 11. MR 0339154

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Additional Information

Dustin Clausen
Affiliation: Department of Mathematics, University of Copenhagen, Copenhagen, Denmark
Email: dustin.clausen@math.ku.dk

Akhil Mathew
Affiliation: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, Massachusetts 02138
Email: amathew@math.harvard.edu

DOI: https://doi.org/10.1090/proc/13648
Received by editor(s): August 10, 2016
Received by editor(s) in revised form: December 16, 2016, and January 2, 2017
Published electronically: June 16, 2017
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2017 American Mathematical Society

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