Strong convergence in the motivic Adams spectral sequence

Abstract

We prove strong convergence results for the motivic Adams spectral sequence of the sphere spectrum over fields with finite virtual cohomological dimension at the prime 2, and over arbitrary fields at odd primes. We show that the motivic Adams spectral sequence is not strongly convergent over number fields. As applications we give bounds on the exponents of the ( , η)-completed motivic stable stems, and calculate the zeroth ( , η)-completed motivic stable stems.CONTENTS 24 9. The zeroth motivic stable stem 24 References 26

Isotropic stable motivic homotopy groups of spheres

“…Strong convergence of the motivic Adams spectral sequence for the sphere spectrum is known also over fields with finite virtual cohomological dimension (see [16]).…”

“…This has been introduced by Morel in [21] and deeply studied by Dugger and Isaksen in [6]. Convergence of the spectral sequence has been analysed by Hu, Kriz and Ormsby in [12] and [13], by Kylling and Wilson in [16] and by Mantovani in [18]. Then, stable homotopy groups of the motivic sphere spectrum have been studied over different base fields by several authors.…”