Inverting the Hopf map

TL;DR

This paper computes the eta-localization of the motivic stable homotopy ring over complex numbers using the motivic Adams-Novikov spectral sequence, confirming a conjecture and verifying proposed differential patterns.

Contribution

It provides the first computation of the eta-localized motivic stable homotopy ring over complex numbers, confirming conjectures and verifying differential patterns in the spectral sequence.

Findings

Confirmed the eta-localization of the motivic stable homotopy ring

Verified differential patterns in the localized motivic Adams spectral sequence

Reduced the problem to classical Adams-Novikov spectral sequence computations

Abstract

We calculate the $\eta$-localization of the motivic stable homotopy ring over the complex numbers, 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 $E_2$-term, and this is what we do. Guillou and Isaksen also proposed a pattern of differentials in the localized motivic classical Adams spectral sequence, which we verify using a method first explored by Novikov.