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We establish duality results for the cohomology of the Weil group of a$p$-adic field, analogous to, but more general than, results from Galoiscohomology. We prove a duality theorem for discrete Weil modules, which impliesTate-Nakayama Duality. We define Weil-smooth cohomology for varieties overlocal fields, and prove a duality theorem for the cohomology of $\G_m$ on asmooth, proper curve with a rational point. This last theorem is analogous to,and implies, a classical duality theorem for such curves.

Weil-\'etale Cohomology over $p$-adic Fields