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

Let $L\mathrm{/}{\mathbb{Q}}_p$ 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 $\mathrm{S}\mathrm{p}\mathrm{f}{\mathcal{O}}_L$ 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 $X=\mathrm{S}\mathrm{p}\mathrm{f}{\mathcal{O}}_K$ for $K\mathrm{/}L$ a $p$-adic field, we show that this recovers the Kisin-Ren equivalence between Lubin-Tate $({\phi }_q,\mathrm{\Gamma })$-modules and ${\mathcal{O}}_L$-linear representations of $G_K$ and the results of Kupferer and Venjakob for computing Galois cohomology in terms of Herr complexes of $({\phi }_q,\mathrm{\Gamma })$-modules. We can thus regard Laurent $F$-crystals on the $L$-typical prismatic site as providing a suitable notion of relative $({\phi }_q,\mathrm{\Gamma })$-modules.