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# Homology, Homotopy and Applications

## Volume 23 (2021)

### Number 1

### The wedge family of the cohomology of the $\mathbb{C}$-motivic Steenrod algebra

Pages: 101 – 117

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a7

#### Author

#### Abstract

We describe some regular behavior in the motivic wedge, which is an infinite family in the cohomology $\mathrm{Ext}_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$ of the $\mathbb{C}$-motivic Steenrod algebra. The key tool is to compare motivic computations to classical computations, to $\mathrm{Ext}_{\mathbf{A}(2)}(\mathbb{M}_2,\mathbb{M}_2)$, or to $h_1$-localization of $\mathrm{Ext}_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$.

We also give two conjectures on the behavior of the families $e_0^tg^k$ and $\Delta h_1 e_0^t g^k$ in $\mathrm{Ext}_{\mathbf{A}}(\mathbb{M}_2,\mathbb{M}_2)$ which raise naturally from the study of the motivic wedge family.

#### Keywords

cohomology of the Steenrod algebra, motivic homotopy theory

#### 2010 Mathematics Subject Classification

55S10, 55T15

Received 5 November 2019

Received revised 20 February 2020

Accepted 3 March 2020

Published 26 August 2020