Perfect even modules and the even filtration

Abstract

Inspired by the work of Hahn-Raksit-Wilson, we introduce a variant of the even filtration which is naturally defined on $\mathbf{E}_{1}$-rings and their modules. We show that our variant satisfies flat descent and so agrees with the Hahn-Raksit-Wilson filtration on ring spectra of arithmetic interest, showing that various "motivic" filtrations are in fact invariants of the $\mathbf{E}_{1}$-structure alone. We prove that our filtration can be calculated via appropriate resolutions in modules and apply it to the study of even cohomology of connective $\mathbf{E}_{1}$-rings, proving vanishing above the Milnor line, base-change formulas, and explicitly calculating cohomology in low weights.