Inverting the Hopf map

Abstract

We calculate the η-localization of the motivic stable homotopy ring over C, confirming a conjecture of Guillou and Isaksen. Our approach is via the motivic Adams-Novikov spectral sequence. In fact, work of Hu, Kriz and Ormsby implies that it suffices to compute the corresponding localization of the classical Adams-Novikov E₂-term, and this is what we do. Guillou and Isaksen also propose a pattern of differentials in the localized motivic classical Adams spectral sequence, which we verify using a method first explored by Novikov.