A short proof of telescopic Tate vanishing

Clausen, Dustin; Mathew, Akhil

doi:10.1090/proc/13648

A short proof of telescopic Tate vanishing
HTML articles powered by AMS MathViewer

by Dustin Clausen and Akhil Mathew

Proc. Amer. Math. Soc. 145 (2017), 5413-5417

DOI: https://doi.org/10.1090/proc/13648

Published electronically: June 16, 2017

PDF | Request permission

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.

References

Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168. MR 2196732, DOI 10.1515/crll.2005.2005.588.149
A. K. Bousfield, On the telescopic homotopy theory of spaces, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391–2426. MR 1814075, DOI 10.1090/S0002-9947-00-02649-0
J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii+178. MR 1230773, DOI 10.1090/memo/0543
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, DOI 10.1007/PL00004264
Michael Hopkins and Jacob Lurie, Ambidexterity in $K(n)$-local stable homotopy theory, available at http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf.
Mark Hovey and Hal Sadofsky, Tate cohomology lowers chromatic Bousfield classes, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3579–3585. MR 1343699, DOI 10.1090/S0002-9939-96-03495-8
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, DOI 10.2307/120991
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 464230, DOI 10.1017/S0305004100054335
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, DOI 10.1007/BFb0085232
Nicholas J. Kuhn, Tate cohomology and periodic localization of polynomial functors, Invent. Math. 157 (2004), no. 2, 345–370. MR 2076926, DOI 10.1007/s00222-003-0354-z
Nicholas J. Kuhn, A guide to telescopic functors, Homology Homotopy Appl. 10 (2008), no. 3, 291–319. MR 2475626
Mark Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. (2) 116 (1982), no. 1, 65–112. MR 662118, DOI 10.2307/2007048
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, DOI 10.1016/0022-4049(81)90064-5
Akhil Mathew, Niko Naumann, and Justin Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017), 994–1084. MR 3570153, DOI 10.1016/j.aim.2016.09.027
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, DOI 10.1112/blms/20.3.262
Graeme Segal, Operations in stable homotopy theory, New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) London Math. Soc. Lecture Note Ser., No. 11, Cambridge Univ. Press, London, 1974, pp. 105–110. MR 0339154