# Localizing the $E_2$ page of the Adams spectral sequence

## TL;DR

This paper investigates the localized structure of the Adams spectral sequence at prime 3, computing differentials up to the E9 page and conjecturing collapse, revealing new insights into the algebraic topology of spheres.

## Contribution

It provides the first detailed computation of the localized Adams E2 page at prime 3, including differential analysis and conjectured spectral sequence collapse.

## Key findings

- Computed up to E9 page of the spectral sequence
- Conjectured spectral sequence collapses at E9
- Complete calculation of localized Ext groups

## Abstract

There is only one nontrivial localization of $\pi_*S_{(p)}$ (the chromatic localization at $v_0=p$), but there are infinitely many nontrivial localizations of the Adams $E_2$ page for the sphere. The first non-nilpotent element in the $E_2$ page after $v_0$ is $b_{10}\in \mathrm{Ext}_A^{2p(p-1)-2}(\mathbb{F}_p,\mathbb{F}_p)$. We work at $p=3$ and study $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ (where $P$ is the algebra of dual reduced powers), which agrees with the infinite summand $\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$ of $\mathrm{Ext}_A(\mathbb{F}_3,\mathbb{F}_3)$ above a line of slope ${1\over 23}$. We compute up to the $E_9$ page of an Adams spectral sequence in the category $\mathrm{Stable}(P)$ converging to $b_{10}^{-1}\mathrm{Ext}_P(\mathbb{F}_3,\mathbb{F}_3)$, and conjecture that the spectral sequence collapses at $E_9$. We also give a complete calculation of $b_{10}^{-1}\mathrm{Ext}_P^*(\mathbb{F}_3,\mathbb{F}_3[\xi_1^3])$.