Motivic $C_p$-power operations in $\mathrm{MGL}$

Abstract

In this paper, we give a new proof of the Hopkins-Morel-Hoyois theorem for fields $k$ of characteristic unequal to $p$ containing all primitive $p$-power roots of unity for a fixed prime $p$. This new proof follows Quillen's proof for the structure of the homotopy groups of $\mathrm{MU}$ which utilizes topological power operations. To this end we define a motivic analog of the Steenrod power operations, using the theory of normed motivic spectra as developed by Bachmann--Hoyois and their application for the definition of motivic colimits as developed by Bachmann-Elmanto-Heller.