On the motivic Segal conjecture

Abstract

We establish motivic versions of the theorems of Lin and Gunawardena, thereby confirming the motivic Segal conjecture for the algebraic group ${\mu }_{\ell }$ of $\ell$th roots of unity, where $\ell$ is any prime. To achieve this we develop motivic Singer constructions associated to the symmetric group $S_{\ell }$ and to ${\mu }_{\ell }$, and introduce a delayed limit Adams spectral sequence.