A stacky comparison of the Hodge and Nygaard filtrations

TL;DR

This paper develops a stack-based framework to compare and generalize filtrations in $p$-adic cohomology theories, extending previous results to broader contexts and introducing new stacky approaches.

Contribution

It formulates a stacky comparison between the Nygaard and Hodge filtrations, generalizes to smooth proper $p$-adic schemes with arbitrary coefficients, and introduces stacky methods for diffracted Hodge cohomology.

Findings

Stacky comparison of filtrations established

Generalization to schemes with arbitrary coefficients achieved

New stack computing conjugate filtration introduced

Abstract

We use the approach to $p$-adic cohomology theories via stacks recently developed by Drinfeld and Bhatt--Lurie to formulate a stacky version of a comparison result between the Nygaard filtration on prismatic cohomology and the Hodge filtration on de Rham cohomology by Bhatt--Lurie and thereby also obtain a generalisation in the case of smooth and proper $p$-adic formal schemes which allows for coefficients in an arbitrary gauge. In the process, we develop a stacky approach to diffracted Hodge cohomology as introduced by Bhatt--Lurie which also captures the conjugate filtration and the Sen operator. In the appendix, we also introduce a stack computing the conjugate filtration on absolute Hodge--Tate cohomology.