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Artin–Verdier duality

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Theorem on constructible abelian sheaves over the spectrum of a ring of algebraic numbers

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

H r ( X , F ) × Ext 3 − r ⁡ ( F , G m ) → H 3 ( X , G m ) = Q / Z {\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(X,F) is the r-th étale cohomology group of the scheme X with values in F, and Ext(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

H r ( X , F ) ∗ ≅ Ext 3 − r ⁡ ( F , G m ) r = 0 , 1 H r ( X , F ) ≅ Ext 3 − r ⁡ ( F , G m ) ∗ r = 2 , 3 {\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

( − ) ∗ = Hom ⁡ ( − , Q / Z ) . {\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

H r ( U , F D ) × H c 3 − r ( U , F ) → H c 3 ( U , G m ) = Q / Z {\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 denotes the Cartier dual of F, which is another finite flat commutative group scheme over U. Moreover, H r ( U , F ) {\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 H c r ( U , F ) {\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

⋯ → H c r ( U , F ) → H r ( U , F ) → ⨁ v ∉ U H r ( K v , F ) → H c r + 1 ( U , F ) → ⋯ {\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(Kv, F) is the Galois cohomology of the Henselization Kv of K at the place v, modified a la Tate:

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

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

References

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