A stacky approach to $p$-adic Hodge theory

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 étale 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 étale cohomology with coefficients in an arbitrary $F$-gauge. This work is the author’s master’s thesis at the University of Bonn.