Artin–Verdier duality explained

In mathematics, Artin–Verdier 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.

Statement

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

H ^{r(X,F) x} \operatorname{Ext} ^{3-r} (F, G _{m)\to} H ^{3 (X, G} _{m) = \Q/\Z}

is a non-degenerate pairing of finite abelian groups, for every integer r.

Here, H^r(X,F) is the r-th étale cohomology group of the scheme X with values in F, and Ext^r(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_m denotes the étale sheaf of units in the structure sheaf of X.

proved Artin–Verdier duality for constructible, but not necessarily torsion sheaves. For such a sheaf F, the above pairing induces isomorphisms

\begin{align} H ^{r(X,} F) ^{*} &\cong \operatorname{Ext} ^{3-r} (F, G _{m)} && r = 0 , 1 \\ H ^{r(X,} F) &\cong \operatorname{Ext} ^{3-r} (F, G ^{*} _{m)} && r = 2 , 3 \end{align}

where

( - ) ^{*} = \operatorname{Hom}( - , \Q /\Z).

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

H ^{r(U,F} ^{D) x} H ^{3- r} _{c} (U,F)\to H ^{3 (U,{ G}} _{m) = \Q/\Z}

of finite abelian groups, for all integers r.

Here F^D denotes the Cartier dual of F, which is another finite flat commutative group scheme over U. Moreover,

H ^{r(U,F)} is the r-th flat cohomology group of the scheme U with values in the flat abelian sheaf F, and H ^{r(U,F)} _{c} 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

… \to H ^{r} _{c(U,F)\to} H ^{r(U,F)\to} o plus\nolimits _{v\notin} H ^{r(K} _{v,F)\to} H ^{r +1} _{c(U,F)} \to …

The sum is taken over all places of K, which are not in U, including the archimedean ones. The local contribution H^r(K_v, F) is the Galois cohomology of the Henselization K_v of K at the place v, modified a la Tate:

H ^{r} _{T( Gal (K} ^{s/K} ^{v),F(K} ^{s)).} ^{v}

Here

K ^{s} _{v} is a separable closure of K _{v.}