Bisynthetic spectra to unify BP^{F_p} and F_p^{BP} deformations

Construct a category of bisynthetic spectra that simultaneously deforms the Adams spectral sequences associated to the BP-synthetic analog of HF_p (denoted BP^{F_p} = ν_{F_p}(BP)) and the HF_p-synthetic analog of BP (denoted F_p^{BP} = ν_{BP}(HF_p)), and show that this unified framework remedies the current failure of the Miller square by yielding a genuine commutative comparison square of spectral sequences tying together the synthetic Adams–Novikov and Cartan–Eilenberg constructions.

Background

The paper constructs a Miller square–type diagram relating the synthetic algebraic Novikov spectral sequence and the motivic Cartan–Eilenberg spectral sequence. However, the authors note that the diagram is not a true Miller square because the relevant objects live in different synthetic categories: S_{F_p} lies in the HF_p-synthetic category, while S_{BP} lies in the BP-synthetic category.

To address this mismatch, the authors propose a bisynthetic category that would deform both BP^{F_p} and F_p^{BP} simultaneously. If such a category existed and had the intended properties, it would provide a unified setting in which the Miller square comparison becomes genuine, enabling direct transfer of differentials and extension computations between the associated spectral sequences.