Kisin, MarkMarks, Samuel2023-06-0220232023-05-152023-05Marks, Samuel. 2023. Laurent F-crystals and Lubin-Tate (φ_q, Γ)-modules. Doctoral dissertation, Harvard University Graduate School of Arts and Sciences.30492666https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37375878Let $L/\Q_p$ be a finite extension. We introduce \textit{$L$-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 $\Spf\O_L$ are equivalent to $\O_L$-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 $X = \Spf\O_K$ for $K/L$ a $p$-adic field, we show that this recovers the Kisin-Ren equivalence between Lubin-Tate $(\varphi_q,\Gamma)$-modules and $\O_L$-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.application/pdfenLubin-Tate formal groupsp-adic Hodge theoryp-divisible groupsPrismatic cohomologyMathematicsThesis or Dissertation2023-06-020009-0002-3965-4389