Localizing the $E_2$ page of the Adams spectral sequence

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 $b_{10}^{-1}{Ext}_P^{\ast ,\ast }\left({\mathbb{𝔽}}_3,{\mathbb{𝔽}}_3\left[{\xi }_1^3\right]\right)$.