The $\mathbb{C}$-motivic wedge subalgebra

TL;DR

This paper investigates the structure of the motivic wedge subalgebra within the $oldsymbol{ ext{Ext}}$ groups of the $oldsymbol{ ext{C}}$-motivic Steenrod algebra, revealing regular patterns and proposing conjectures based on comparisons to classical and localized computations.

Contribution

It introduces a new perspective on the motivic wedge by comparing motivic and classical Ext computations and formulates a conjecture on specific families within the motivic Ext.

Findings

Identification of regular patterns in the motivic wedge

Comparison techniques linking motivic and classical Ext computations

A conjecture on the behavior of the family $e_0^tg^k$ in motivic Ext

Abstract

We describe some regular behavior in the motivic wedge, which is a subalgebra of the cohomology Ext$_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$ of the $\mathbb{C}$-motivic Steenrod algebra. The key tool is to compare motivic computations to classical computations, to Ext$_{\mathbf{A}(2)}(\mathbb{M}_2,\mathbb{M}_2)$, or to $h_1$-localization of Ext$_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$. We also give a conjecture on the behavior of the family $e_0^tg^k$ in Ext$_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$ which raises naturally from the study of the motivic wedge.