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 \operatorname{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=\operatorname{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} , as well as 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.