Prismatic F-crystals and E-crystalline Galois representations

2023/06/22 by Pham, Dat
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2306.13205

openalex publication_date 2023/06/22 · openalex created_date 2023/06/27 · openalex updated_date 2026/07/28

Abstract

Let K be a complete discretely valued field of mixed characteristic (0,p) with perfect residue field, and let E be a finite extension of Q_p contained in K. We show that the category of prismatic F-crystals on O_K (relative to E in a suitable sense) is equivalent to the category of O_E-lattices in E-crystalline representations defined by Kisin--Ren, extending the main result of \citearxiv:2106.14735 in the case E=Q_p. As a key ingredient in the proof, by adapting a lemma of Du--Liu, we prove a general full faithfulness result for certain vector bundles on the prismatic site, which simplifies and refines the key descent step in the approach of Bhatt--Scholze without invoking the Beilinson fibre sequence.

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