AasTRACT. We discuss the first author's PiC&l'd groups of stable homotopy. We give 1I. detailed description of the calculation of Pie" and go on to describe geometric constructions for lifts of the elements of PiCl' We also construct a 15 ceU complex that localizes to what we speculate ill an interesting element of Pic 2. For all n we describe an algebraic approxima.tion to Picn using the Ada.ms-Novikov spectral sequence. We also show that the p-adic integers embed in the group Pic n for all n and p. 1991 Mu.thematic~Subject ClaBificalion. Primary 55P42; Seconda.ry 55QI0. The authors were parLially supporLed by the National Science Founda.tion. This article is in final form and no version of it will be subm.itted for pubiicalion elsewhere.

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We construct splittings of some completions of the Z/(p)-Tate cohomology of E(n) and some related spectra. In particular, we split (a completion of) tE(n) as a (completion of) a wedge of E(n − 1)'s as a spectrum, where t is shorthand for the fixed points of the Z/(p)-Tate cohomology spectrum (ie the Mahowald inverse limit lim. We also give a multiplicative splitting of tE(n) after a suitable base extension.

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