Artin–Verdier Duality

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

:$H^r(X,F)\times \operatorname^(F,\mathbb_m)\to H^3(X,\mathbb_m)=\Q/\Z$

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

:$\begin H^r(X, F)^* &\cong \operatorname^(F, \mathbb_m) && r = 0, 1 \\ H^r(X, F) &\cong \operatorname^(F, \mathbb_m)^* && r = 2, 3 \end$

where

:$(-)^* = \operatorname(-, \Q /\Z).$

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

:$H^r(U,F^D)\times H_c^(U,F)\to H_c^3(U,_m)=\Q/\Z$

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 $H_c^r(U,F)$ 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

:$\cdots\to H^r_c(U,F)\to H^r(U,F)\to \bigoplus\nolimits_ H^r(K_v,F)\to H^_c(U,F) \to\cdots$

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:

:$H^r(K_v,F)=H^r_T(\mathrm(K_v^s/K_v),F(K_v^s)).$

Here $K_v^s$ is a separable closure of $K_v.$

References

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{{DEFAULTSORT:Artin-Verdier duality
Theorems in number theory
Duality theories