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Duality for $p$-adic geometric pro-étale cohomology I: a Fargues-Fontaine avatar (2411.12163v1)
Published 19 Nov 2024 in math.AG and math.NT
Abstract: $p$-adic geometric pro-\'etale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields can be represented by solid quasi-coherent sheaves on the Fargues-Fontaine curve. We prove that these sheaves satisfy a Poincar\'e duality. This is done by passing, via comparison theorems, to analogous sheaves representing syntomic cohomology and then reducing to Poincar\'e duality for ${\mathbf B}+_{\rm st}$-twisted Hyodo-Kato and filtered ${\mathbf B}+_{\rm dR}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result. A similar computation yields a K\"unneth formula.