Prismatic $F$-crystals and $E$-crystalline Galois representations

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 $\mathbf{Q}_p$ contained in $K$. We show that the category of prismatic $F$-crystals on $\mathcal{O}_K$ (relative to $E$ in a suitable sense) is equivalent to the category of $\mathcal{O}_E$-lattices in $E$-crystalline representations defined by Kisin--Ren, extending the main result of \cite{arxiv:2106.14735} in the case $E=\mathbf{Q}_p$. 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.