Hodge--Tate prismatic crystals and Sen theory

TL;DR

This paper develops a classification of Hodge-Tate crystals on the prismatic site of a mixed characteristic ring, introduces a Sen theory over a Kummer tower, and connects these to nearly Hodge-Tate representations, with several cohomology results.

Contribution

It introduces a new classification of Hodge-Tate crystals via modules with endomorphisms and constructs a Sen theory over a non-Galois Kummer tower, linking crystals to nearly Hodge-Tate representations.

Findings

Classified Hodge-Tate crystals by modules with endomorphisms.

Constructed Sen theory over a non-Galois Kummer tower.

Proved cohomology comparison and vanishing results.

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.