The Dual Motivic Witt Cohomology Steenrod Algebra

TL;DR

This paper investigates the structure of the dual Steenrod algebra in motivic Witt cohomology, revealing algebraic properties over certain fields and connecting to recent computations of related spectra.

Contribution

It determines the algebra structure of the dual Steenrod algebra for motivic Witt cohomology over specific fields, extending understanding in motivic homotopy theory.

Findings

Computed the algebra structure of ${H_W ext{Z}}_{**}H_W ext{Z}$ over certain fields.

Inverted $ ext{η}$ to analyze the algebra ${HW}_{**}HW$ and related spectra.

Connected results to recent work by Bachmann and Hopkins on $HW$-module structures.

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 ${H_W\mathbb{Z}}_{**}H_W\mathbb{Z}$ over fields $k$ of characteristic not $2$ which are extensions of fields $F$ with $K^M_2(F)/2=0$. 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 ${HW}_{**}HW$. In particular, for the given base fields, this implies the $HW$-module structure of $HW\wedge HW$ recently computed by Bachmann and Hopkins.