Some quotient Hopf algebras of the dual Steenrod algebra
Author:
J. H. Palmieri
Journal:
Trans. Amer. Math. Soc. 358 (2006), 671685
MSC (2000):
Primary 55S10, 18G15, 20E18, 20J06
Published electronically:
March 10, 2005
MathSciNet review:
2177035
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Fix a prime , and let be the polynomial part of the dual Steenrod algebra. The Frobenius map on induces the Steenrod operation on cohomology, and in this paper, we investigate this operation. We point out that if , then for any element in the cohomology of , if one applies 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 in which the Frobenius is an isomorphism of order . We show that these quotients are dual to group algebras, the resulting groups are torsionfree, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about . The dual complete Steenrod algebra makes an appearance.
 [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
(2001i:55020), http://dx.doi.org/10.1007/BF02803522
 [Cam00]
Rachel
Camina, The Nottingham group, New horizons in pro𝑝
groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA,
2000, pp. 205–221. MR 1765121
(2001f:20054)
 [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 (97i:55032)
 [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
(43 #6915)
 [Mil58]
John
Milnor, The Steenrod algebra and its dual, Ann. of Math. (2)
67 (1958), 150–171. MR 0099653
(20 #6092)
 [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
(52 #6711)
 [Pal99]
John
H. Palmieri, Quillen stratification for the Steenrod algebra,
Ann. of Math. (2) 149 (1999), no. 2, 421–449.
MR
1689334 (2000g:55026), http://dx.doi.org/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
(2002a:55019), http://dx.doi.org/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
(45 #7743)
 [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
(87j:55003)
 [Ser02]
JeanPierre
Serre, Galois cohomology, Corrected reprint of the 1997
English edition, Springer Monographs in Mathematics, SpringerVerlag,
Berlin, 2002. Translated from the French by Patrick Ion and revised by the
author. MR
1867431 (2002i:12004)
 [Arn94]
 D. Arnon, Generalized Dickson invariants, Ph.D. thesis, Mass. Inst. of Tech., 1994.
 [Arn00]
 , Generalized Dickson invariants, Israel J. Math. 118 (2000), 183205. MR 2001i:55020
 [Cam00]
 R. Camina, The Nottingham group, New horizons in pro groups, Birkhäuser Boston, Boston, MA, 2000, pp. 205221. MR 2001f:20054
 [LH96]
 I. Llerena and N. 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), Birkhäuser, Basel, 1996, pp. 271284. MR 97i:55032
 [May70]
 J. P. 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) (F. P. Peterson, ed.), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 153231. MR 43:6915
 [Mil58]
 J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150171. MR 20:6092
 [Nak75]
 O. Nakamura, Some differentials in the Adams spectral sequence, Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. (1975), no. 19, 125. MR 52:6711
 [Pal99]
 J. H. Palmieri, Quillen stratification for the Steenrod algebra, Ann. of Math. (2) 149 (1999), no. 2, 421449. MR 2000g:55026
 [Pal01]
 J. H. Palmieri, Stable homotopy over the Steenrod algebra, Mem. Amer. Math. Soc. 151 (2001), no. 716, xiv+172. MR 2002a:55019
 [Qui71]
 D. G. Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549572, 573602. MR 45:7743
 [Rav86]
 D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press Inc., Orlando, FL, 1986. MR 87j:55003
 [Ser02]
 J.P. Serre, Galois cohomology, English ed., Springer Monographs in Mathematics, SpringerVerlag, Berlin, 2002, Translated from the French by Patrick Ion and revised by the author. MR 2002i:12004
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 981954350
Email:
palmieri@math.washington.edu
DOI:
http://dx.doi.org/10.1090/S0002994705036378
PII:
S 00029947(05)036378
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
