Prismatic $F$-crystals and Lubin-Tate $(\varphi_q,\Gamma)$-modules

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 \'etale 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 $(\varphi_q,\Gamma)$-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 $(\varphi_q,\Gamma)$-modules. We can thus regard Laurent $F$-crystals on the $L$-typical prismatic site as providing a suitable notion of relative $(\varphi_q,\Gamma)$-modules.