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

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 $B_{\mathrm{S}\mathrm{e}\mathrm{n}}$ in a geometric way.