An arithmetic étale-crystalline comparison with coefficients in crystalline local systems
Maximilian Hauck
TL;DR
The paper develops a stack-theoretic framework for p-adic cohomology theories to generalize the crystalline–étale comparison with coefficients in crystalline local systems. Central to the approach are the constructions of the prismatic/de Rham/Nygaard stacks $X^{ m dR}$, $X^{oldsymbol{ riangle}}$, $X^{ m N}$, and the syntomification $X^{ m Syn}$, which furnish universal coefficient theories via $F$-gauges. A stacky Beilinson fibre square with coefficients is established, yielding a natural comparison between syntomic and étale cohomology with coefficients and connecting to Fontaine–Messing syntomic cohomology; the results extend Colmez–Nizioł’s work to arbitrary crystalline local systems. The paper also identifies an isogeny category of perfect $F$-gauges on $ ext{Z}_p$ and proves a crystalline realization of étale representations, unifying multiple realizations (de Rham, crystalline, étale) in a single geometric framework. Collectively, these results provide a robust, integral perspective on p-adic motives and extend the reach of stacky and prismatic methods in p-adic Hodge theory, with concrete implications for comparing cohomology theories with coefficient systems.
Abstract
We use the stacky approach to $p$-adic cohomology theories recently developed by Drinfeld and Bhatt--Lurie to generalise a comparison theorem 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 $X$ due to Colmez--Niziol to the case of coefficients in an arbitrary crystalline local system on the generic fibre of $X$. In the process, we 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. Our methods also yield a description of the isogeny category of perfect $F$-gauges on $\mathbb{Z}_p$.