Arithmetic duality for p -adic pro-étale cohomology of analytic curves

Abstract We prove a Poincaré duality for arithmetic p -adic pro-étale cohomology of smooth dagger curves over finite extensions of bold upper Q Subscript p Q p $\mathbf{Q}_p$ . We deduce it, via the Hochschild–Serre spectral sequence, from geometric comparison theorems combined with Tate and Serre dualities. The compatibility of all the products involved is checked via reduction to the ghost circle, for which we also prove a Poincaré duality (showing that it behaves like a proper smooth analytic variety of dimension 1 divided by 2 1 / 2 $1/2$ ). Along the way we study functional analytic properties of arithmetic p -adic pro-étale cohomology and prove that the usual cohomology is nuclear Fréchet and the compactly supported one: of compact type.