Some quotient Hopf algebras of the dual Steenrod algebra
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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
- D. Arnon, Generalized Dickson invariants, Ph.D. thesis, Mass. Inst. of Tech., 1994.
- Dan Arnon, Generalized Dickson invariants, Israel J. Math. 118 (2000), 183–205. MR 1776082, DOI 10.1007/BF02803522
- 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
- Irene Llerena and NguyĂŞn H. V. Hu’ng, The complete Steenrod algebra and the generalized Dickson algebra, Algebraic topology: new trends in localization and periodicity (Sant Feliu de GuĂxols, 1994) Progr. Math., vol. 136, Birkhäuser, Basel, 1996, pp. 271–284. MR 1397738
- 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
- John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171. MR 99653, DOI 10.2307/1969932
- Osamu Nakamura, Some differentials in the $\textrm {mod}\ 3$ Adams spectral sequence, Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. 19 (1975), 1–25. MR 0385852
- John H. Palmieri, Quillen stratification for the Steenrod algebra, Ann. of Math. (2) 149 (1999), no. 2, 421–449. MR 1689334, DOI 10.2307/120969
- John H. Palmieri, Stable homotopy over the Steenrod algebra, Mem. Amer. Math. Soc. 151 (2001), no. 716, xiv+172. MR 1821838, DOI 10.1090/memo/0716
- 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 298694, DOI 10.2307/1970770
- 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
- 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
Additional Information
- J. H. Palmieri
- Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
- Email: palmieri@math.washington.edu
- Received by editor(s): January 7, 2003
- Received by editor(s) in revised form: February 19, 2004
- Published electronically: March 10, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 671-685
- MSC (2000): Primary 55S10, 18G15, 20E18, 20J06
- DOI: https://doi.org/10.1090/S0002-9947-05-03637-8
- MathSciNet review: 2177035