On Artin-Verdier duality for function fields
Let X be a smooth, complete curve over a finite field and F a constructible sheaf on the étale site \(U_{et}\) of a nonempty, open subset \(U\subset X\). In this note the author proves that, for all \(i\in {\mathbb{Z}}\), the Yoneda pairing \[ H^ i_ c(U,F)\times Ext_ U^{3- i}(F,{\mathbb{G}}_ m)\to H^ 3_ c(U,{\mathbb{G}}_ m)\overset \sim \rightarrow^{tr}{\mathbb{Q}}\quad /{\mathbb{Z}} \] is a nondegenerate pairing of finite groups, and that the trace map tr (induced by the degree map) is an isomorphism. If \(X=Spec({\mathfrak O}_ K)\) where \({\mathfrak O}_ K\) is the ring of integers of a totally imaginary number field, then the analogous statement is known as the Artin-Verdier duality theorem; cf. \textit{M. Artin}, \textit{J. L. Verdier} [Sémin. on étale cohomologie of number fields, Woods Hole 1964] and \textit{B. Mazur} [Ann. Sci. École Norm. Supér., IV. Sér. 6(1973), 521-556 (1974; Zbl 0282.14004); the author remarks that in the latter paper there is a gap in the argument, which he fills here.] Similar duality theorems (for flat cohomology) were given by \textit{M. Artin} and \textit{J. Milne} [Invent. Math. 35, 111-129 (1976; Zbl 0342.14007)] and \textit{G. Frey} [J. Reine Angew. Math. 301, 46-58 (1978; Zbl 0375.12015)] In recent papers the author extended the above result in two directions: (1) to \({\mathbb{Z}}\)-constructible sheaves [J. Reine Angew. Math. 366, 18-31 (1986)]; and (2) to singular curves [''Duality in the étale cohomology of one-dimensional proper schemes and generalizations'' (preprint)].
scientific article; zbMATH DE number 3937316
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