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On the $ BP\langle n\rangle$-cohomology of elementary abelian $ p$-groups


Author: Geoffrey Powell
Journal: Trans. Amer. Math. Soc. 368 (2016), 8029-8046
MSC (2010): Primary 55N20, 55N22, 20J06
DOI: https://doi.org/10.1090/tran6699
Published electronically: January 27, 2016
MathSciNet review: 3546792
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Abstract | References | Similar Articles | Additional Information

Abstract: The structure of the $ BP\langle n\rangle $-cohomology of elementary abelian
$ p$-groups is studied, obtaining a presentation expressed in terms of $ BP$-
cohomology and mod-$ p$ singular cohomology, using the Milnor derivations.

The arguments are based on a result on multi-Koszul complexes which is related to Margolis's criterion for freeness of a graded module over an exterior algebra.


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  • [Boa95] J. Michael Boardman, Stable operations in generalized cohomology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 585-686. MR 1361899 (97b:55021), https://doi.org/10.1016/B978-044481779-2/50015-8
  • [CK89] David Carlisle and Nicholas J. Kuhn, Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras, J. Algebra 121 (1989), no. 2, 370-387. MR 992772 (90c:55018), https://doi.org/10.1016/0021-8693(89)90073-2
  • [Har91] Shin-ichiro Hara, The Hopf rings for connective Morava $ K$-theory and connective complex $ K$-theory, J. Math. Kyoto Univ. 31 (1991), no. 1, 43-70. MR 1093326 (92c:55005)
  • [HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, 553-594 (electronic). MR 1758754 (2001k:55015), https://doi.org/10.1090/S0894-0347-00-00332-5
  • [JW73] David Copeland Johnson and W. Stephen Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973), 327-353. MR 0334257 (48 #12576)
  • [JW85] David Copeland Johnson and W. Stephen Wilson, The Brown-Peterson homology of elementary $ p$-groups, Amer. J. Math. 107 (1985), no. 2, 427-453. MR 784291 (86j:55008), https://doi.org/10.2307/2374422
  • [JWY94] David Copeland Johnson, W. Stephen Wilson, and Dung Yung Yan, Brown-Peterson homology of elementary $ p$-groups. II, Topology Appl. 59 (1994), no. 2, 117-136. MR 1296028 (95j:55008), https://doi.org/10.1016/0166-8641(94)90090-6
  • [JY80] David Copeland Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms, Osaka J. Math. 17 (1980), no. 1, 117-136. MR 558323 (81b:55010)
  • [Kan82] Richard Kane, Finite $ H$-spaces and the Thom map for connective $ K$-theory, Current trends in algebraic topology, Part 1 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 389-415. MR 686127 (84b:55016)
  • [Kuh87] Nicholas J. Kuhn, The Morava $ K$-theories of some classifying spaces, Trans. Amer. Math. Soc. 304 (1987), no. 1, 193-205. MR 906812 (89d:55013), https://doi.org/10.2307/2000710
  • [Lan92] Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'un $ p$-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135-244 (French). With an appendix by Michel Zisman. MR 1179079 (93j:55019)
  • [Mar83] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland Mathematical Library, vol. 29, North-Holland Publishing Co., Amsterdam, 1983. Modules over the Steenrod algebra and the stable homotopy category. MR 738973 (86j:55001)
  • [Pow14] Geoffrey M. L. Powell, On connective $ K$-theory of elementary abelian 2-groups and local duality, Homology Homotopy Appl. 16 (2014), no. 1, 215-243. MR 3211744, https://doi.org/10.4310/HHA.2014.v16.n1.a13
  • [RWY98] Douglas C. Ravenel, W. Stephen Wilson, and Nobuaki Yagita, Brown-Peterson cohomology from Morava $ K$-theory, $ K$-Theory 15 (1998), no. 2, 147-199. MR 1648284 (2000d:55012), https://doi.org/10.1023/A:1007776725714
  • [Str00] N. P. Strickland, The $ {\rm BP}\langle n\rangle $ cohomology of elementary abelian groups, J. London Math. Soc. (2) 61 (2000), no. 1, 93-109. MR 1745398 (2001j:55006), https://doi.org/10.1112/S0024610799008340
  • [Tam97] Hirotaka Tamanoi, The image of the BP Thom map for Eilenberg-Mac Lane spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1209-1237. MR 1401530 (97i:55012), https://doi.org/10.1090/S0002-9947-97-01826-6
  • [Tam00] Hirotaka Tamanoi, Spectra of BP-linear relations, $ v_n$-series, and BP cohomology of Eilenberg-Mac Lane spaces, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5139-5178. MR 1661270 (2001b:55023), https://doi.org/10.1090/S0002-9947-99-02484-8
  • [Wil73] W. Stephen Wilson, The $ \Omega $-spectrum for Brown-Peterson cohomology. I, Comment. Math. Helv. 48 (1973), 45-55; corrigendum, ibid. 48 (1973), 194. MR 0326712 (48 #5055)
  • [Wil75] W. Stephen Wilson, The $ \Omega $-spectrum for Brown-Peterson cohomology. II, Amer. J. Math. 97 (1975), 101-123. MR 0383390 (52 #4271)
  • [Wil84] W. Stephen Wilson, The Hopf ring for Morava $ K$-theory, Publ. Res. Inst. Math. Sci. 20 (1984), no. 5, 1025-1036. MR 764345 (86c:55008), https://doi.org/10.2977/prims/1195180879

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Additional Information

Geoffrey Powell
Affiliation: Laboratoire Angevin de Recherche en Mathématiques, UMR 6093, Faculté des Sciences, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers, France
Email: Geoffrey.Powell@math.cnrs.fr

DOI: https://doi.org/10.1090/tran6699
Keywords: Brown-Peterson theory, Johnson-Wilson theories, Milnor primitive, elementary abelian group, formal group
Received by editor(s): November 20, 2013
Received by editor(s) in revised form: January 13, 2015, and March 13, 2015
Published electronically: January 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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