Hopf Rings

The category of graded, bicommutative Hopf algebras over the prime eld 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 comes down to computing E 2 S 3 { which Ravenel has done for E = BP. From this one recovers the work of Hopkins, Hunton, and Turner on the Hopf rings of Landweber exact cohomology theories. The are two major algebraic diiculties encountered in this approach. The rst 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 Bousseld: torsion-free Hopf algebras over the p-adic integers with some additional structure, such as a self-Hopf-algebra map that reduces to the Verschiebung, can be easily classiied. An abelian category A with a set of small projective generators is equivalent to a category M of modules over some ring R. In addition, if A and M are symmetric monoidal categories and the equivalence of categories A ! M respects the monoidal structure, one can study the ring objects in A by studying the ring objects in M. The purpose of this paper is to develop this observation in the case where A is the category HA + of graded, bicommutative Hopf algebras over the prime eld F p. The graded ring objects in HA + are called Hopf rings and they arise naturally when studying unstable cohomology operations for some cohomology theory E (see 2, 11, 21]). To state some results, x a prime p > 2. (A slight rewording gives the results at p = 2). We will restrict attention to the sort of Hopf algebra that arises in algebraic topology; …