Poincaré duality for pro-étale ${\mathbf{Q}}_p$-local systems

We prove finiteness and Poincaré duality for pro-étale ${\mathbf{Q}}_p$-local systems on proper $p$-adic rigid-analytic spaces in the framework of Banach–Colmez spaces and establish their optimal cohomological vanishing bounds. As a consequence, we also obtain ordinary finiteness and duality for their arithmetic pro-étale cohomology. We deduce these results from their analogs for perfect complexes on the Fargues–Fontaine curve, which in turn reduce to Poincaré duality for perfect complexes over period rings. We give a simple proof of the latter duality via the same diagrammatic argument as in our previous work for finite coefficients. Along the way, we establish optimal $v$-descent for perfect complexes over certain period rings on affinoid perfectoid spaces.