v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack

TL;DR

This paper offers a geometric perspective on Sen theory by describing continuous semilinear Galois representations over p-adic fields using vector bundles on the Hodge-Tate stack, connecting cohomology and period rings.

Contribution

It introduces a novel geometric framework for understanding Sen theory through vector bundles on the Hodge-Tate locus, linking Galois representations and period rings.

Findings

Describes the category of continuous semilinear representations via vector bundles.

Provides a geometric explanation for the appearance of Colmez' period ring in Sen theory.

Connects Galois cohomology with the geometry of the Hodge-Tate stack.

Abstract

We describe the category of continuous semilinear representations and their cohomology for the Galois group of a $p$-adic field $K$ with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring $B_{\mathrm{Sen}}$ in a geometric way.