On the Hopf Ring for a Symplectic Oriented Spectrum

1. Introduction Given any graded commutative ring R with identity, a Hopf ring over R is a com-mutative ring in the category of graded R-coalgebras. Such objects arise naturally in algebraic topology, as explained below, and their investigation was pioneered by Ravenel and Wilson in 14]. To any spectrum G we may associate an-spectrum fG r : r 2 Zg by choosing G r to be the space representing the cohomology functor G r (). Given a second spectrum E, we may then consider the bigraded homology module E (G). If we assume that both spectra are multiplicative and that the homology functor E () behaves suitably well with respect to products, then E (G) constitutes a Hopf ring over the coeecients E. Ravenel and Wilson concentrated on the cases when both G and E are complex oriented , as deened in 1]. In this situation each of the corresponding cohomology theories gives rise to a formal group law, and the interaction of these two laws forms a central pillar of their theory. The universal complex oriented example is the complex bordism spectrum MU, and they succesfully analysed the Hopf ring E (MU) for several diierent E. Subsequently, other Hopf rings have been investigated by similar methods, as in 8] and 15], but little is known about global structure in the absence of complex orientability. However, if we restrict our attention to the nonnegatively graded spaces fG r : r 0g, written as G 0 , the homology module is still a Hopf ring and the picture becomes clearer for certain other important spectra. For example, if we let S denote the sphere spectrum (which is initial in the category of all spectra, and for which S r is more usually written QS r), and if we take E to be the mod p Eilenberg-MacLane spectrum for some prime p, then E (S 0) has recently been described purely in terms of the Dyer-Lashof algebra acting on the canonical zero-dimensional class 22]. The corresponding action in the case of MU has also been determined 9], 21]. These observations suggest that the global Hopf ring machinery is good for computation with complex orientable examples whose coeecient rings are known, but that for cases which are closer to the sphere spectrum it is more fruitful to consider the nonnegatively graded homology Hopf ring in terms of the Dyer-Lashof operations. It is therefore …