Prismatic F-crystals and E-crystalline Galois representations

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 Ø K (relative to E in a suitable sense) is equivalent to the category of Ø E -lattices in E-crystalline G K -representations introduced by Kisin–Ren, extending a previous result of Bhatt–Scholze 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.