Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Some quotient Hopf algebras of the dual Steenrod algebra


Author: J. H. Palmieri
Journal: Trans. Amer. Math. Soc. 358 (2006), 671-685
MSC (2000): Primary 55S10, 18G15, 20E18, 20J06
Published electronically: March 10, 2005
MathSciNet review: 2177035
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Fix a prime $p$, and let $A$ be the polynomial part of the dual Steenrod algebra. The Frobenius map on $A$ induces the Steenrod operation $\widetilde{\mathscr{P}}^{0}$on cohomology, and in this paper, we investigate this operation. We point out that if $p=2$, then for any element in the cohomology of $A$, if one applies $\widetilde{\mathscr{P}}^{0}$ enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that ``enough times'' should be ``once.''

The bulk of the paper is a study of some quotients of $A$ in which the Frobenius is an isomorphism of order $n$. We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about $\widetilde{\mathscr{P}}^{0}$. The dual complete Steenrod algebra makes an appearance.


References [Enhancements On Off] (What's this?)

  • [Arn94] D. Arnon, Generalized Dickson invariants, Ph.D. thesis, Mass. Inst. of Tech., 1994.
  • [Arn00] Dan Arnon, Generalized Dickson invariants, Israel J. Math. 118 (2000), 183–205. MR 1776082, 10.1007/BF02803522
  • [Cam00] Rachel Camina, The Nottingham group, New horizons in pro-𝑝 groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 205–221. MR 1765121
  • [LH96] Irene Llerena and Nguyên H. V. Hu’ng, The complete Steenrod algebra and the generalized Dickson algebra, Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994) Progr. Math., vol. 136, Birkhäuser, Basel, 1996, pp. 271–284. MR 1397738
  • [May70] J. Peter May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153–231. MR 0281196
  • [Mil58] John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 0099653
  • [Nak75] Osamu Nakamura, Some differentials in the 𝑚𝑜𝑑3 Adams spectral sequence, Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. 19 (1975), 1–25. MR 0385852
  • [Pal99] John H. Palmieri, Quillen stratification for the Steenrod algebra, Ann. of Math. (2) 149 (1999), no. 2, 421–449. MR 1689334, 10.2307/120969
  • [Pal01] John H. Palmieri, Stable homotopy over the Steenrod algebra, Mem. Amer. Math. Soc. 151 (2001), no. 716, xiv+172. MR 1821838, 10.1090/memo/0716
  • [Qui71] Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 0298694
  • [Rav86] Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
  • [Ser02] Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR 1867431

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55S10, 18G15, 20E18, 20J06

Retrieve articles in all journals with MSC (2000): 55S10, 18G15, 20E18, 20J06


Additional Information

J. H. Palmieri
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
Email: palmieri@math.washington.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03637-8
Received by editor(s): January 7, 2003
Received by editor(s) in revised form: February 19, 2004
Published electronically: March 10, 2005
Article copyright: © Copyright 2005 American Mathematical Society