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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Tate cohomology lowers chromatic
Bousfield classes


Authors: Mark Hovey and Hal Sadofsky
Journal: Proc. Amer. Math. Soc. 124 (1996), 3579-3585
MSC (1991): Primary 55P60, 55P42; Secondary 55N91
MathSciNet review: 1343699
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Abstract: Let $G$ be a finite group. We use recent results of J. P. C. Greenlees and H. Sadofsky to show that the Tate homology of $E(n)$ local spectra with respect to $G$ produces $E(n-1)$ local spectra. We also show that the Bousfield class of the Tate homology of $L_{n}X$ (for $X$ finite) is the same as that of $L_{n-1}X$. To be precise, recall that Tate homology is a functor from $G$-spectra to $G$-spectra. To produce a functor $P_{G}$ from spectra to spectra, we look at a spectrum as a naive $G$-spectrum on which $G$ acts trivially, apply Tate homology, and take $G$-fixed points. This composite is the functor we shall actually study, and we'll prove that $\langle P_{G}(L_{n}X) \rangle = \langle L_{n-1}X \rangle $ when $X$ is finite. When $G = \Sigma _{p}$, the symmetric group on $p$ letters, this is related to a conjecture of Hopkins and Mahowald (usually framed in terms of Mahowald's functor ${\mathbf R}P_{-\infty }(-)$).


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Additional Information

Mark Hovey
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: hovey@math.mit.edu

Hal Sadofsky
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21230
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: hs@math.jhu.edu, sadofsky@math.uoregon.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03495-8
Received by editor(s): January 19, 1995
Received by editor(s) in revised form: June 8, 1995
Additional Notes: The authors were partially supported by the NSF
Communicated by: Tom Goodwillie
Article copyright: © Copyright 1996 American Mathematical Society