Geodesic
Prismatic $F$-crystals and Lubin–Tate $(\varphi_{q},\Gamma)$-modules
Documenta mathematica, Tome 30 (2025) no. 4, pp. 787-838

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Let L/Qp​ 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 SpfOL​ are equivalent to OL​-linear local systems on the generic fiber Xη​. 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=SpfOK​ for K/L a p-adic field, we show that this recovers the Kisin–Ren equivalence between Lubin–Tate (φq​,Γ)-modules and OL​-linear representations of GK​, as well as the results of Kupferer and Venjakob for computing Galois cohomology in terms of Herr complexes of (φq​,Γ)-modules. We can thus regard Laurent F-crystals on the L-typical prismatic site as providing a suitable notion of relative (φq​,Γ)-modules.
DOI : 10.4171/dm/1009
Classification : 14F30, 11F85, 11S31, 11S20
Mots-clés : p-adic Hodge theory, prismatic cohomology, (φ,Γ)-modules, Lubin–Tate formal groups 
@article{10_4171_dm_1009,
     author = {Samuel Marks},
     title = {Prismatic $F$-crystals and {Lubin{\textendash}Tate} $(\varphi_{q},\Gamma)$-modules},
     journal = {Documenta mathematica},
     pages = {787--838},
     publisher = {mathdoc},
     volume = {30},
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     year = {2025},
     doi = {10.4171/dm/1009},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/1009/}
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Samuel Marks. Prismatic $F$-crystals and Lubin–Tate $(\varphi_{q},\Gamma)$-modules. Documenta mathematica, Tome 30 (2025) no. 4, pp. 787-838. doi: 10.4171/dm/1009

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