Here, Hr(X,F) is the r-th
étale cohomology group of the
schemeX with values in F, and Extr(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, 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
Finite flat group schemes
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. 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 Hr(Kv, F) is the
Galois cohomology of the
HenselizationKv of K at the place v, modified a la
Tate: