Some quotient Hopf algebras of the dual Steenrod algebra
by J. H. Palmieri
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 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 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 P^0. The dual
complete Steenrod algebra makes an appearance.