Abstract. Resolutions, which generalize the classical Koszul resolutions, are constructed for a large class of augmented algebras including the Steenrod algebra and the universal enveloping algebras.

No abstract

1. Introduction. This paper will show that after localization at any given primep, the infinite loop space structure on the space BSO is essentially unique. If the word 'localization' is replaced by 'completion', the result continues to hold; and both results continue to hold if the space BSO is replaced by the space B8U.In order to state this result formally, it is natural to suppose given a connected Q-spectrum X whose Oth term X o is equivalent to the localization or completion at p of BG, where G = SO or SU according to the case. One should then state and prove that this spectrum X is equivalent to some fixed spectrum Y. Provided we arrange for Y to be an O-spectrum, this conclusion shows that there is an equivalence of infinite loop spaces from X o to the fixed infinite loop space Y o .Our proof that X ~ Y falls into two parts. The first step determines the modp cohomology of X as a module over the modp Steenrod algebra. The second step starts from a knowledge of the mod p cohomology of X, and constructs an equivalence of spectra X -> Y.The second step is valid not only for the cases G = SO and G = SU, but also for the cases G = 0 and G = U; but in the latter cases the first step is not vah'd in the form we have discussed so far. That is, in these cases, we require information about X o not only as a space, but as an .ff-space.We therefore begin formal work by considering the second step, and for this purpose we first construct the ' obvious' fixed spectrum Y.Let K R be the spectrum which represents classical (periodic) real K-theory; similarly for Kc in the complex case. Let d be a fixed integer; let bg be the spectrum obtained from K E or Kc by killing homotopy groups in degrees < d, while retaining the homotopy groups in degrees > d. The spectrum bg therefore represents (d -l)-connected isT-theory (real or complex). The notation bg is chosen to reflect the usual notation for connective isT-theory; one obtains bo and bso from K E by taking d = 1 and d = 2, while one obtains bu and bsu from K c by taking d = 2 and d = 4.Let p be a fixed prime. Let A be either the ring Z p of j?-adic integers, or the ring Z (2) ) of integers localized &tp (that is, the ring of fractions a/b with a, b integers and b prime top). We can introduce coefficients A into any spectrum W by setting W A = MA A W, where MA is a Moore spectrum for the group A. We take our fixed spectrum Y to be

No abstract

No abstract