The motivic Adams spectral sequence

TL;DR

This paper explores the structure and convergence of the motivic Adams spectral sequence over algebraically closed fields, introduces motivic analogues of classical spectral sequences, and uncovers new motivic homotopy classes without classical counterparts.

Contribution

It provides new insights into the motivic Adams spectral sequence, including its construction, convergence, and applications to classical results and exotic motivic classes.

Findings

Data on motivic Steenrod algebra cohomology

Motivic versions of May and Adams-Novikov spectral sequences

Existence of exotic motivic homotopy classes

Abstract

We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field. We discuss several features of the associated Adams spectral sequence, including the basic construction and convergence properties. The paper also deals with motivic versions of the May and Adams-Novikov spectral sequences. It is shown how these tools can be used to give new proofs of some classical results in algebraic topology. Also, the considerations reveal the existence of certain "exotic" motivic homotopy classes which have no classical analogues.