Hodge--Tate prismatic crystals and Sen theory

Abstract

We study Hodge-Tate crystals on the absolute (log-) prismatic site of ${\mathcal{O}}_K$, where ${\mathcal{O}}_K$ is a mixed characteristic complete discrete valuation ring with perfect residue field. We first classify Hodge-Tate crystals by ${\mathcal{O}}_K$-modules equipped with certain small endomorphisms. We then construct Sen theory over a non-Galois Kummer tower, and use it to classify rational Hodge-Tate crystals by (log-) nearly Hodge-Tate representations. Various cohomology comparison and vanishing results are proved along the way.

v1:This preprint contains almost all results of arXiv:2112.10140 (except section 5 there) and all results of arXiv:2201.10136 . Furthermore, all results are generalized to the log-prismatic setting. Incidentally, this preprint has a same title as arxiv:2201.10136; we apologize for possible confusions. v2: improved exposition, very minor revisions. Comments are welcome