Localizing the E2 page of the Adams spectral sequence

Belmont, Eva

doi:10.2140/agt.2020.20.1965

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Localizing the $E_2$ page of the Adams spectral sequence

Eva Belmont

Algebraic & Geometric Topology 20 (2020) 1965–2028

DOI: 10.2140/agt.2020.20.1965

Abstract

There is only one nontrivial localization of $π_{*} S_{(p)}$ (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 nonnilpotent element in the $E_{2}$ page after $v_{0}$ is $b_{10} \in {Ext}_{A}^{2, 2 p (p - 1)} (𝔽_{p}, 𝔽_{p})$ . We work at $p = 3$ and study $b_{10}^{- 1} {Ext}_{P}^{*, *} (𝔽_{3}, 𝔽_{3})$ (where $P$ is the algebra of dual reduced powers), which agrees with the infinite summand ${Ext}_{P}^{*, *} (𝔽_{3}, 𝔽_{3})$ of ${Ext}_{A}^{*, *} (𝔽_{3}, 𝔽_{3})$ above a line of slope $\frac{1}{23}$ . We compute up to the $E_{9}$ page of an Adams spectral sequence in the category $Stable (P)$ converging to $b_{10}^{- 1} {Ext}_{P}^{*, *} (𝔽_{3}, 𝔽_{3})$ , and conjecture that the spectral sequence collapses at $E_{9}$ . We also give a complete calculation of $b_{10}^{- 1} {Ext}_{P}^{*, *} (𝔽_{3}, 𝔽_{3} [ξ_{1}^{3}])$ .

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Keywords

Adams spectral sequence, localized Ext, Cartan–Eilenberg spectral sequence, Ivanovskii spectral sequence, Margolis–Palmieri Adams spectral sequence

Mathematical Subject Classification 2010

Primary: 55T15

References

Publication

Received: 28 January 2019

Revised: 8 September 2019

Accepted: 13 November 2019

Published: 20 July 2020

Authors

Eva Belmont
Department of Mathematics Northwestern University Evanston, IL United States

Volume 20, issue 4 (2020)