We discuss the Adams Spectral Sequence for R-modules based on commutative localized regular quotient ring spectra over a commutative S -algebra R in the sense of Elmendorf, Kriz, Mandell, May and Strickland. The formulation of this spectral sequence is similar to the classical case and the calculation of its E 2 -term involves the cohomology of certain 'brave new Hopf algebroids' E R * E . In working out the details we resurrect Adams' original approach to Universal Coefficient Spectral Sequences for modules over an R ring spectrum.We show that the Adams Spectral Sequence for S R based on a commutative localized regular quotient R ring spectrum E = R/I[X −1 ] converges to the homotopy of the E -nilpotent completionWe also show that when the generating regular sequence of I * is finite,the Bousfield localization of S R with respect to E -theory. The spectral sequence here collapses at its E 2 -term but it does not have a vanishing line because of the presence of polynomial generators of positive cohomological degree. Thus only one of Bousfield's two standard convergence criteria applies here even though we have this equivalence. The details involve the construction of an I -adic towerwhose homotopy limit is L R E S R . We describe some examples for the motivating case R = M U .

It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions Symm. We offer the cohomology of the space CP ∞ as a topological model for the ring of quasisymmetric functions QSymm. We exploit standard results from topology to shed light on some of the algebraic properties of QSymm. In particular, we reprove the Ditters conjecture. We investigate a product on CP ∞ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU . The canonical Thom spectrum over CP ∞ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.

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Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum at a prime p. They discussed May's notions of nuclear complexes and of cores of spaces, spectra, and commutative S-algebras. Their most striking conclusions, due to Hu and Kriz, were negative: cores are not unique up to equivalence, and BP is not a core of M U considered as a commutative S-algebra, although it is a core of M U considered as a p-local spectrum. We investigate these ideas further, obtaining much more positive conclusions. We show that nuclear complexes have several non-obviously equivalent characterizations. Up to equivalence, they are precisely the irreducible complexes, the minimal atomic complexes, and the Hurewicz complexes with trivial mod p Hurewicz homomorphism above the Hurewicz dimension, which we call complexes with no mod p detectable homotopy. Unlike the notion of a nuclear complex, these other notions are all invariant under equivalence. This simple and conceptual criterion for a complex to be minimal atomic allows us to prove that many familiar spectra, such as ko, eo 2 , and BoP at the prime 2, all BP n at any prime p, and the indecomposable wedge summands of Σ ∞ CP ∞ and Σ ∞ HP ∞ at any prime p are minimal atomic.1991 Mathematics Subject Classification. Primary 55P15, 55P42, 55P60. Key words and phrases. atomic space, atomic spectrum, nuclear space, nuclear spectrum. L CJ 2n−1Since J 2n−1 is a wedge of (2n − 1)-spheres and π 2n−1 (F p) = 0, [J 2n−1 , F p] = 0. A standard result, given in just this form in [14, Lemma 1], shows that there are maps f n+1 and h n+1 that make the diagram commute. Passing to colimits, we obtain f and a homotopy h : q ≃ p • f n . Since the homology groups of BoP are concentrated in even degrees [15], we can replace it by a minimal complex, with cells only in even degrees. This allows us to reverse the roles of X and BoP to construct g.A similar argument proves the following result.