Poincar\'e duality for pro-\'etale $\mathbf{Q}_p$-local systems

Abstract

We prove finiteness and Poincar\'e duality for pro-\'etale $\mathbf{Q}_p$-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-\'etale cohomology. We deduce these results from their analogs for perfect complexes on the Fargues--Fontaine curve, which in turn reduce to Poincar\'e 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.