Spectral weight filtrations

TL;DR

This paper describes a new approach to motivic spectra using weight filtrations on cohomologies, enabling effective calculations in positive characteristic and refining structures in complex cases.

Contribution

It introduces a novel description of motivic spectra via motives of smooth proper varieties and constructs compatible weight filtrations on various cohomologies.

Findings

Constructed weight filtrations on Betti and étale cohomologies.

Proved these filtrations satisfy h-descent, facilitating calculations in positive characteristic.

Refined weight filtrations at the level of stable homotopy types for complex motivic cases.

Abstract

We provide a description of Voevodsky's $\infty$-category of motivic spectra in terms of the subcategory of motives of smooth proper varieties. As applications, we construct weight filtrations on the Betti and \'{e}tale cohomologies of algebraic varieties with coefficients in any complex oriented ring spectrum. We show that these filtrations satisfy $\ell\mathrm{dh}$-descent, giving an effective way of calculating them in positive characteristic. In the complex motivic case, we further refine the weight filtration to one defined at the level of stable homotopy types.