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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, weshow 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 localsystems on the generic fiber $X_\eta$. We also give comparison theorems forcomputing the \'etale cohomology of a local system in terms of the cohomologyof its corresponding Laurent $F$-crystal. In the case $X =\mathrm{Spf}\mathcal{O}_K$ for $K/L$ a $p$-adic field, we show that thisrecovers 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 cohomologyin terms of Herr complexes of $(\varphi_q,\Gamma)$-modules. We can thus regardLaurent $F$-crystals on the $L$-typical prismatic site as providing a suitablenotion of relative $(\varphi_q,\Gamma)$-modules.

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