Prismatic $F$-crystals and Lubin–Tate $({\phi }_q,\Gamma )$-modules

Abstract

Let $L/{\mathbb{Q}}_p$ be a finite extension. We introduce $L$-typical prisms, a mild generalization of prisms. Following ideas of Bhatt, Scholze, and Wu, we show that certain vector bundles, called Laurent $F$-crystals, on the $L$-typical prismatic site of a formal scheme $X$ over $\mathrm{Spf}{\mathcal{O}}_L$ are equivalent to ${\mathcal{O}}_L$-linear local systems on the generic fiber $X_{\eta }$. We also give comparison theorems for computing the étale cohomology of a local system in terms of the cohomology of its corresponding Laurent $F$-crystal. In the case $X=\mathrm{Spf}{\mathcal{O}}_K$ for $K/L$ a $p$-adic field, we show that this recovers the Kisin–Ren equivalence between Lubin–Tate $({\phi }_q,\Gamma )$-modules and ${\mathcal{O}}_L$-linear representations of $G_K$, as well as the results of Kupferer and Venjakob for computing Galois cohomology in terms of Herr complexes of $({\phi }_q,\Gamma )$-modules. We can thus regard Laurent $F$-crystals on the $L$-typical prismatic site as providing a suitable notion of relative $({\phi }_q,\Gamma )$-modules.