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.
Moreover, Gm denotes the étale sheaf of units in the structure sheaf of X. Christopher Deninger (1986) proved Artin–Verdier 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 FD denotes the Cartier dual of F, which is another finite flat commutative group scheme over U.