The Dual Motivic Witt Cohomology Steenrod Algebra

Abstract

In this paper we begin the study of the (dual) Steenrod algebra of the motivic Witt cohomology spectrum $H_W\mathbb{Z}$ by determining the algebra structure of ${H_W\mathbb{Z}}_{\ast \ast }{H}_W\mathbb{Z}$ over fields $k$ of characteristic not $2$ which are extensions of fields $F$ with $K_2^M(F)/2=0$. For example, this includes all fields of odd characteristic, as well as fields that are extensions of quadratically closed fields of characteristic $0$. After inverting $\eta$, this computes the $HW:=H_W\mathbb{Z}[{\eta }^{-1}]$-algebra ${HW}_{\ast \ast }HW$. In particular, for the given base fields, this implies the $HW$-module structure of $HW\wedge HW$ recently computed by Bachmann and Hopkins.

22 pages