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We describe the category of continuous semilinear representations and theircohomology for the Galois group of a $p$-adic field $K$ with coefficients in acompleted algebraic closure via vector bundles on the Hodge-Tate locus of theCartier-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.

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