  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.
