Conjecture on mixed Bockstein differentials and τ-periodicity
Establish the equivalence asserted in Conjecture Type3diffs between (i) the existence of a τ^{2^n}-periodic differential d_t(τ^{2^n}x)=ρ^t z in the ℝ-motivic ρ-Bockstein spectral sequence for a τ-free class x with d_r(x)=ρ^r y where y is τ-power torsion and r<t<2^n, and (ii) the existence of the nonperiodic differential d_{t−r}(Q/ρ^{t−r} y)=γ/τ^{2^n} z in the ρ-Bockstein spectral sequence for the negative cone; moreover, deduce the predicted τ^{2^n}-extension from y to ρ^{t−r}z in Ext_ℝ(F_2/ρ^{t+1}).
References
Suppose that x and y are classes in \Ext_\mathbb{R} such that x is \tau-free and y is \tau-power torsion. Suppose furthermore that d_r(x)= \rhor y in the \mathbb{R}-motivic \rho-Bockstein spectral sequence and that r < t < 2n. There exists a \tau{2n}-periodic differential d_t(\tau{2n}x) = \rhot z if and only if the (nonperiodic) differential d_{t-r} \left( \frac{Q}{\rho{t-r}} y \right) = \frac{\gamma}{\tau{2n}} z
occurs in the \rho-Bockstein spectral sequence for the negative cone.
If these differentials occur, then there is a \tau{2n}-extension from y to \rho{t-r} z in \Ext_\mathbb{R} \left( \frac{\mathbb{F}_2}{\rho{t+1}} \right) that is hidden by the \rho-Bockstein spectral sequence.