In mathematics, ArtinâÂÂVerdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by , that generalizes Tate duality.
It shows that, as far as etale (or flat) cohomology is concerned, the ring of integers in a number field behaves like a 3-dimensional mathematical object.
Let X be the spectrum of the ring of integers in a totally imaginary number field K, and F a constructible étale abelian sheaf on X. Then the Yoneda pairing
is a non-degenerate pairing of finite abelian groups, for every integer r.
Here, H<sup>r</sup>(X,F) is the r-th étale cohomology group of the scheme X with values in F, and Ext<sup>r</sup>(F,G) is the group of r-extensions of the étale sheaf G by the étale sheaf F in the category of étale abelian sheaves on X. Moreover, G<sub>m</sub> denotes the étale sheaf of units in the structure sheaf of X.
proved ArtinâÂÂVerdier duality for constructible, but not necessarily torsion sheaves. For such a sheaf F, the above pairing induces isomorphisms
where
Let U be an open subscheme of the spectrum of the ring of integers in a number field K, and F a finite flat commutative group scheme over U. Then the cup product defines a non-degenerate pairing
of finite abelian groups, for all integers r.
Here F<sup>D</sup> denotes the Cartier dual of F, which is another finite flat commutative group scheme over U. Moreover, is the r-th flat cohomology group of the scheme U with values in the flat abelian sheaf F, and is the r-th flat cohomology with compact supports of U with values in the flat abelian sheaf F.
The flat cohomology with compact supports is defined to give rise to a long exact sequence
The sum is taken over all places of K, which are not in U, including the archimedean ones. The local contribution H<sup>r</sup>(K<sub>v</sub>, F) is the Galois cohomology of the Henselization K<sub>v</sub> of K at the place v, modified a la Tate:
Here is a separable closure of