The motivic Adams spectral sequence

Abstract

We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field of characteristic 0. Our results are based on computer calculations and a motivic version of the May spectral sequence. We discuss features of the associated Adams spectral sequence and use these tools to give new proofs of some results in classical algebraic topology. We also consider a motivic AdamsNovikov spectral sequence. The investigations reveal the existence of some stable motivic homotopy classes that have no classical analogue. 55T15, 14F42

Remarks on motivic homotopy theory over algebraically closed fields

“…Acknowledgements: Since the first version of this paper was posted on the Ktheory archive, the paper [8], the results of which partially (but not fully) overlap with ours, appeared on arXiv. We thank Dan Dugger for a subsequent discussion, which prompted us to make some clarifications in the present version.…”

The η-local motivic sphere

“…We recall the details of the motivic May spectral sequence from [6]. This spectral sequence has four gradings: three from Ext A and one additional May grading associated with the filtration involved in construction of the spectral sequence.…”

“…Working over C (or any algebraically closed field of characteristic zero), we have an Adams spectral sequence for computing motivic stable 2-complete homotopy groups with good convergence properties [12,6,7]. The E 2 -page of this spectral sequence is the cohomology Ext A of the motivic Steenrod algebra A.…”

Convergence of the Motivic Adams Spectral Sequence

“…In the following sections, by the statement that "the motivic Adams spectral sequence converges for X" we shall simply mean that X → X Ad is an equivalence. For essentially formal reasons, the Adams spectral sequence always converges to the homotopy groups of X Ad , [5], Section 6, also [6].…”