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.