Colmez–Gilles–Nizioł geometric duality for pro-étale cohomology

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Let KK be a finite extension of Qp{\bf Q}_p, let YY be a smooth partially proper rigid analytic variety over KK of dimension dd, and let Rproeˊt⁡,∗(YC,Qp(r)){\mathbb R}_{\operatorname{pro\acute{e}t},*}(Y_C,{\bf Q}_p(r)) denote the TVS-valued complexes representing pro-étale cohomology and its compactly supported variant. Geometric duality conjecture. There is a natural duality quasi-isomorphism in the TVS category

Rproeˊt⁡(YC,Qp)≃RHomTVS(Rproeˊt⁡,c(YC,Qp(d))[2d],Qp).{\mathbb R}_{\operatorname{pro\acute{e}t}}(Y_C,{\bf Q}_p)\simeq {\rm R}{\cal Hom}_{\rm TVS}({\mathbb R}_{\operatorname{pro\acute{e}t},c}(Y_C,{\bf Q}_p(d))[2d],{\bf Q}_p).

The source states that this conjecture is now a theorem, with proofs by Anschütz–Le Bras–Mann and Colmez–Gilles–Nizioł.

References

Primary source

Pierre Colmez and Wiesława Nizioł, “Hodge Theory of p-adic analytic varieties: a survey”, arXiv:2601.16557 (2026).

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