Localizing the e2 page of the adams spectral sequence

There is only one nontrivial localization of π∗S(p) (the chromatic localization at v0=p), but there are infinitely many nontrivial localizations of the Adams E2 page for the sphere. The first nonnilpotent element in the E2 page after v0 is b10∈ ExtA2,2p(p−1)(𝔽p,𝔽p). We work at p=3 and study b10−1 ExtP∗,∗(𝔽3,𝔽3) (where P is the algebra of dual reduced powers), which agrees with the infinite summand ExtP∗,∗(𝔽3,𝔽3) of ExtA∗,∗(𝔽3,𝔽3) above a line of slope 123. We compute up to the E9 page of an Adams spectral sequence in the category Stable(P) converging to b10−1 ExtP∗,∗(𝔽3,𝔽3), and conjecture that the spectral sequence collapses at E9. We also give a complete calculation of b10−1 ExtP∗,∗(𝔽3,𝔽3[ξ13]).