Dyadic Steenrod algebra and its applications

TL;DR

This paper extends Steenrod operations to dyadic integers, introduces a new algebraic framework, and connects it with rigid analytic geometry, providing novel tools for the Peterson hit problem.

Contribution

It develops the dyadic Steenrod algebra over $\\mathbb{Z}_2$, introduces integration-like operations, and links the algebra to rigid analytic geometry and Tate algebras.

Findings

Dyadic Steenrod squares $Jq^k$ extend classical operations.

The dyadic Steenrod algebra $\\mathcal{J}_2$ is an Ore domain allowing localization.

A new norm characterizes hit elements in formal power series.

Abstract

First, by inspiration of the results of Wood \cite{differential,problems}, but with the methods of non-commutative geometry and different approach, we extend the coefficients of the Steenrod squaring operations from the filed $\mathbb{F}_2$ to the dyadic integers $\mathbb{Z}_2$ and call the resulted operations the dyadic Steenrod squares, denoted by $Jq^k$. The derivation-like operations $Jq^k$ generate a graded algebra, called the dyadic Steenrod algebra, denoted by $\mathcal{J}_2$ acting on the polynomials $\mathbb{Z}_2[\xi_1, \dots, \xi_n]$. Being $\mathcal{J}_2$ an Ore domain, enable us to localize $\mathcal{J}_2$ which leads to the appearance of the integration-like operations $Jq^{-k}$ satisfying the $Jq^{-k}Jq^k=1=Jq^kJq^{-k}$. These operations are enough to exhibit a kind of differential equation, the dyadic Steenrod ordinary differential equation. Then we prove that the completion of $\mathbb{Z}_2[\xi_1, \dots, \xi_n]$ in the linear transformation norm coincides with a certain Tate algebra. Therefore, the rigid analytic geometry is closely related to the dyadic Steenrod algebra. Finally, we define the Adem norm $\| \ \|_A$ in which the completion of $\mathbb{Z}_2[\xi_1, \dots, \xi_n]$ is $\mathbb{Z}_2\llbracket\xi_1,\dots,\xi_n\rrbracket$, the $n$-variable formal power series. We surprisingly prove that an element $f \in \mathbb{Z}_2\llbracket \xi_1,\dots,\xi_n\rrbracket$ is hit if and only if $\|f\|_A<1$. This suggests new techniques for the traditional Peterson hit problem in finding the bases for the cohit modules.