A note on \'etale $(\varphi,\Gamma)$-modules in families

TL;DR

This paper establishes an equivalence between prismatic $(\Lambda,F)$-crystals and $\Lambda$-étale local systems on the generic fiber of a formal scheme, connecting prismatic cohomology with pro-étale cohomology and Galois representations.

Contribution

It introduces a new equivalence between prismatic crystals and étale local systems, and links prismatic cohomology with Iwasawa cohomology and Galois representations.

Findings

Equivalence between prismatic $(\Lambda,F)$-crystals and étale local systems.

Recovery of pro-étale cohomology from prismatic cohomology.

Reproof of Dee's classical Galois representation equivalence.

Abstract

Let $\Lambda$ be a complete noetherian local ring with finite residue field of characteristic $p$ and $K/\mathbb{Q}_p$ a $p$-adic field. We show that, by deformation of the structure sheaf on the (transversal) prismatic site of a bounded $p$-adic formal scheme $\mathfrak{X}$, the category of prismatic $(\Lambda,F)$-crystals on $\mathfrak{X}$ is equivalent to $\Lambda$-\'etale local systems on the generic adic fiber of $\mathfrak{X}$ and that the cohomology of $(\Lambda,F)$-crystals recovers the pro-\'etale cohomology of the corresponding local systems. The proof follows the strategy used in \cite{bhatt2023prismatic} and \cite{marks2023prismatic}. From this we construct an isomorphism between Iwasawa cohomology of a $p$-adic Lie extension of $K$ and prismatic cohomology. Following \cite{wu2021galois}, we then reprove Dee's classical result \cite{article} on the equivalence between families of Galois representations and \'etale $(\varphi,\Gamma)$-modules.