A stacky approach to $p$-adic Hodge theory

TL;DR

This paper extends $p$-adic Hodge theory comparison theorems using a stacky approach, allowing for coefficients in various local systems and filtrations, thus broadening the scope of existing results.

Contribution

It generalizes comparison theorems in $p$-adic Hodge theory to include coefficients and develops a stacky framework for diffracted Hodge and syntomic cohomology.

Findings

Established comparison between rational crystalline and $p$-adic étale cohomology with coefficients.

Proved the compatibility of Nygaard and Hodge filtrations for arbitrary gauges.

Developed a stacky approach to diffracted Hodge cohomology and syntomic cohomology with coefficients.

Abstract

We use the stacky approach to $p$-adic cohomology theories recently developed by Drinfeld and Bhatt--Lurie to generalise known comparison theorems in $p$-adic Hodge theory so as to accommodate coefficients. More precisely, we establish a comparison between the rational crystalline cohomology of the special fibre and the rational $p$-adic \'etale cohomology of the arithmetic generic fibre of any proper $p$-adic formal scheme $X$ which allows for coefficients in any crystalline local system on the generic fibre of $X$; moreover, we also prove a comparison between the Nygaard filtration and the Hodge filtration for coefficients in an arbitrary gauge in the sense of Bhatt--Lurie. In the process, we develop a stacky approach to diffracted Hodge cohomology as introduced by Bhatt--Lurie, establish a version of the Beilinson fibre square of Antieau--Mathew--Morrow--Nikolaus with coefficients in the proper case and prove a comparison between syntomic cohomology and $p$-adic \'etale cohomology with coefficients in an arbitrary $F$-gauge. This work is the author's master's thesis at the University of Bonn.