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 Michael Artin and Jean-Louis Verdier (1964), 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

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

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.

Christopher Deninger (1986) proved Artin–Verdier duality for constructible, but not necessarily torsion sheaves. For such a sheaf F, the above pairing induces isomorphisms

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

where

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

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

${\displaystyle H^{r}(U,F^{D})\times H_{c}^{3-r}(U,F)\to H_{c}^{3}(U,{\mathbb {G} }_{m})=\mathbb {Q} /\mathbb {Z} }$ 

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

${\displaystyle \cdots \to H_{c}^{r}(U,F)\to H^{r}(U,F)\to \bigoplus \nolimits _{v\notin U}H^{r}(K_{v},F)\to H_{c}^{r+1}(U,F)\to \cdots }$ 

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:

${\displaystyle H^{r}(K_{v},F)=H_{T}^{r}(\mathrm {Gal} (K_{v}^{s}/K_{v}),F(K_{v}^{s})).}$ 

Here ${\displaystyle K_{v}^{s}}$  is a separable closure of ${\displaystyle K_{v}.}$ 

References

• Artin, Michael; Verdier, Jean-Louis (1964), "Seminar on étale cohomology of number fields", (PDF), Providence, R.I.: American Mathematical Society, archived from the original (PDF) on 2011-05-26
• Deninger, Christopher (1986), "An extension of Artin-Verdier duality to nontorsion sheaves", Journal für die reine und angewandte Mathematik, 366: 18–31, doi:10.1515/crll.1986.366.18, MR 0833011
• Mazur, Barry (1973), "Notes on étale cohomology of number fields", Annales Scientifiques de l'École Normale Supérieure, Série 4, 6: 521–552, ISSN 0012-9593, MR 0344254
• Milne, James S. (2006), Arithmetic duality theorems (Second ed.), BookSurge, LLC, pp. viii+339, ISBN 1-4196-4274-X