John H. Palmieri

STABLY THICK SUBCATEGORIES OF MODULES OVER HOPF ALGEBRAS

We discuss a general method for classifying certain subcategories of the category of nite-dimensional modules over a nite-dimensional cocom- mutative Hopf algebra B. Our method is based on that of Benson-Carlson- Rickard (BCR96), who classify such subcategories when B = kG, the group ring of a nite group G over an algebraically closed eld k. We get a similar classication when B is a nite sub-Hopf algebra of the mod 2 Steenrod alge- bra, with scalars extended to the algebraic closure of F2 .A long the way, we prove a Quillen stratication theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

Some quotient Hopf algebras of the dual Steenrod algebra

Fix a prime p, and let A be the polynomial part of the dual Steen- rod algebra. The Frobenius map on A induces the Steenrod operation f 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 f 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 f P 0 . The dual complete Steenrod algebra makes an appearance.

Algebraic Topology (Hanoi, August 2004): The problem session

This article contains a collection of problems contributed during the course of the conference.