Title: Hopf rings, Dieudonn\'e modules, and $E_\ast \Omega^2S^3$.
Author: Paul Goerss
AMS Classification Nos: 55205, 55N20, 57T05, 16W30
Department of Mathematics
Northwestern University
Evanston IL 60208
pgoerss@math.washington.edu (until 8/24/98)
pogerss@math.nwu.edu
(01/19/99)
This is a highly revised version of the original submission (7/27/99).
P.G.
Abstract: Hopf algebras over the prime field with $p$ elements is an abelian
category which is equivalent, by work of Schoeller, to a category of
graded modules, known as Dieudonn\'e modules. Graded ring objects in Hopf
algebras are called Hopf rings, and they arise in the study of unstable
cohomology operations for extraordinary cohomology theories. The central
point of this paper is that Hopf rings can be studied by looking at the
associated ring object in Dieudonn\'e modules. They can also be computed
there, and because of the relationship between Brown-Gitler spectra and
Dieudonn\'e modules, calculating the Hopf ring for a homology theory $E_\ast$
comes down to computing $E_\ast\Omega^2S^3$ -- which Ravenel has done for
$E = BP$.
The are two major algebraic difficulties encountered in this approach.
The first is to decide what a ring object is in the category of Dieudonn\'e
modules, as there is no obvious symmetric monoidal pairing associated to a
tensor product of modules. The second is to show that Hopf rings pass to
rings in Dieudonn\'e modules. This involves studying universal examples, and
here we pick up an idea suggested by Bousfield: torsion-free Hopf
algebras over the $p$-adic integers with some additional structure,
such as a self-Hopf-algebra map that reduces to the Frobenius, can be
easily classified.