#### Volume 20, issue 4 (2020)

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Localizing the $E_2$ page of the Adams spectral sequence

### Eva Belmont

Algebraic & Geometric Topology 20 (2020) 1965–2028
##### Abstract

There is only one nontrivial localization of ${\pi }_{\ast }{S}_{\left(p\right)}$ (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 nonnilpotent element in the ${E}_{2}$ page after ${v}_{0}$ is ${b}_{10}\in {Ext}_{A}^{2,2p\left(p-1\right)}\left({\mathbb{𝔽}}_{p},{\mathbb{𝔽}}_{p}\right)$. We work at $p=3$ and study ${b}_{10}^{-1}{Ext}_{P}^{\ast ,\ast }\left({\mathbb{𝔽}}_{3},{\mathbb{𝔽}}_{3}\right)$ (where $P$ is the algebra of dual reduced powers), which agrees with the infinite summand ${Ext}_{P}^{\ast ,\ast }\left({\mathbb{𝔽}}_{3},{\mathbb{𝔽}}_{3}\right)$ of ${Ext}_{A}^{\ast ,\ast }\left({\mathbb{𝔽}}_{3},{\mathbb{𝔽}}_{3}\right)$ above a line of slope $\frac{1}{23}$. We compute up to the ${E}_{9}$ page of an Adams spectral sequence in the category $Stable\left(P\right)$ converging to ${b}_{10}^{-1}{Ext}_{P}^{\ast ,\ast }\left({\mathbb{𝔽}}_{3},{\mathbb{𝔽}}_{3}\right)$, and conjecture that the spectral sequence collapses at ${E}_{9}$. We also give a complete calculation of ${b}_{10}^{-1}{Ext}_{P}^{\ast ,\ast }\left({\mathbb{𝔽}}_{3},{\mathbb{𝔽}}_{3}\left[{\xi }_{1}^{3}\right]\right)$.

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##### Keywords
Adams spectral sequence, localized Ext, Cartan–Eilenberg spectral sequence, Ivanovskii spectral sequence, Margolis–Palmieri Adams spectral sequence
Primary: 55T15
##### Publication
Revised: 8 September 2019
Accepted: 13 November 2019
Published: 20 July 2020
##### Authors
 Eva Belmont Department of Mathematics Northwestern University Evanston, IL United States