Idempotents, Localizations and Picard groups of A(1)-modules

We analyze the stable isomorphism type of polynomial rings on degree 1 generators as modules over the sub-algebra A(1) = <Sq^1, Sq^2> of the mod 2 Steenrod algebra. Since their augmentation ideals are Q_1-local, we do this by studying the Q_i-local subcategories and the associated Margolis localizations. The periodicity exhibited by such modules reduces the calculation to one that is finite. We show that these are the only localizations which preserve tensor products, by first computing the Picard groups of these subcategories and using them to determine all idempotents in the stable category of bounded-below A(1)-modules. We show that the Picard groups of the whole category are detected in the local Picard groups, and show that every bounded-below A(1) -module is uniquely expressible as an extension of a Q_0-local module by a Q_1-local module, up to stable equivalence. Applications include correct, complete proofs of Ossa's theorem, applications to Powell's work describing connective K-theory of classifying spaces of elementary abelian groups in functorial terms, and Ault's work on the hit problem.