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Tate duality

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In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by John Tate (1962) and Georges Poitou (1967).

Local Tate duality

Main article: local Tate duality

For a p-adic local field k {\displaystyle k} , local Tate duality says there is a perfect pairing of the finite groups arising from Galois cohomology:

H r ( k , M ) × H 2 − r ( k , M ′ ) → H 2 ( k , G m ) = Q / Z {\displaystyle \displaystyle H^{r}(k,M)\times H^{2-r}(k,M')\rightarrow H^{2}(k,\mathbb {G} _{m})=\mathbb {Q} /\mathbb {Z} }

where M {\displaystyle M} is a finite group scheme, M ′ {\displaystyle M'} its dual Hom ⁡ ( M , G m ) {\displaystyle \operatorname {Hom} (M,G_{m})} , and G m {\displaystyle \mathbb {G} _{m}} is the multiplicative group. For a local field of characteristic p > 0 {\displaystyle p>0} , the statement is similar, except that the pairing takes values in H 2 ( k , μ ) = ⋃ p ∤ n 1 n Z / Z {\displaystyle H^{2}(k,\mu )=\bigcup _{p\nmid n}{\tfrac {1}{n}}\mathbb {Z} /\mathbb {Z} } . The statement also holds when k {\displaystyle k} is an Archimedean field, though the definition of the cohomology groups looks somewhat different in this case.

Global Tate duality

Given a finite group scheme M {\displaystyle M} over a global field k {\displaystyle k} , global Tate duality relates the cohomology of M {\displaystyle M} with that of M ′ = Hom ⁡ ( M , G m ) {\displaystyle M'=\operatorname {Hom} (M,G_{m})} using the local pairings constructed above. This is done via the localization maps

α r , M : H r ( k , M ) → ∏ v ′ H r ( k v , M ) , {\displaystyle \alpha _{r,M}:H^{r}(k,M)\rightarrow {\prod _{v}}'H^{r}(k_{v},M),}

where v {\displaystyle v} varies over all places of k {\displaystyle k} , and where ∏ ′ {\displaystyle \prod '} denotes a restricted product with respect to the unramified cohomology groups. Summing the local pairings gives a canonical perfect pairing

∏ v ′ H r ( k v , M ) × ∏ v ′ H 2 − r ( k v , M ′ ) → Q / Z . {\displaystyle {\prod _{v}}'H^{r}(k_{v},M)\times {\prod _{v}}'H^{2-r}(k_{v},M')\rightarrow \mathbb {Q} /\mathbb {Z} .}

One part of Poitou-Tate duality states that, under this pairing, the image of H r ( k , M ) {\displaystyle H^{r}(k,M)} has annihilator equal to the image of H 2 − r ( k , M ′ ) {\displaystyle H^{2-r}(k,M')} for r = 0 , 1 , 2 {\displaystyle r=0,1,2} .

The map α r , M {\displaystyle \alpha _{r,M}} has a finite kernel for all r {\displaystyle r} , and Tate also constructs a canonical perfect pairing

ker ( α 1 , M ) × ker ⁡ ( α 2 , M ′ ) → Q / Z . {\displaystyle {\text{ker}}(\alpha _{1,M})\times \ker(\alpha _{2,M'})\rightarrow \mathbb {Q} /\mathbb {Z} .}

These dualities are often presented in the form of a nine-term exact sequence

0 → H 0 ( k , M ) → ∏ v ′ H 0 ( k v , M ) → H 2 ( k , M ′ ) ∗ {\displaystyle 0\rightarrow H^{0}(k,M)\rightarrow {\prod _{v}}'H^{0}(k_{v},M)\rightarrow H^{2}(k,M')^{*}}
→ H 1 ( k , M ) → ∏ v ′ H 1 ( k v , M ) → H 1 ( k , M ′ ) ∗ {\displaystyle \rightarrow H^{1}(k,M)\rightarrow {\prod _{v}}'H^{1}(k_{v},M)\rightarrow H^{1}(k,M')^{*}}
→ H 2 ( k , M ) → ∏ v ′ H 2 ( k v , M ) → H 0 ( k , M ′ ) ∗ → 0. {\displaystyle \rightarrow H^{2}(k,M)\rightarrow {\prod _{v}}'H^{2}(k_{v},M)\rightarrow H^{0}(k,M')^{*}\rightarrow 0.}

Here, the asterisk denotes the Pontryagin dual of a given locally compact abelian group.

All of these statements were presented by Tate in a more general form depending on a set of places S {\displaystyle S} of k {\displaystyle k} , with the above statements being the form of his theorems for the case where S {\displaystyle S} contains all places of k {\displaystyle k} . For the more general result, see e.g. Neukirch, Schmidt & Wingberg (2000, Theorem 8.4.4).

Poitou–Tate duality

Among other statements, Poitou–Tate duality establishes a perfect pairing between certain Shafarevich groups. Given a global field k {\displaystyle k} , a set S of primes, and the maximal extension k S {\displaystyle k_{S}} which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of Gal ⁡ ( k S / k ) {\displaystyle \operatorname {Gal} (k_{S}/k)} which vanish in the Galois cohomology of the local fields pertaining to the primes in S.

An extension to the case where the ring of S-integers O S {\displaystyle {\mathcal {O}}_{S}} is replaced by a regular scheme of finite type over Spec ⁡ O S {\displaystyle \operatorname {Spec} {\mathcal {O}}_{S}} was shown by Geisser & Schmidt (2018). Another generalisation is due to Česnavičius, who relaxed the condition on the localising set S by using flat cohomology on smooth proper curves.

See also

References

  1. Neukirch, Schmidt & Wingberg (2000, Theorem 7.2.6)
  2. See Neukirch, Schmidt & Wingberg (2000, Theorem 8.6.8) for a precise statement.
  3. Česnavičius, Kęstutis (2015). "Poitou–Tate without restrictions on the order" (PDF). Mathematical Research Letters. 22 (6): 1621–1666. doi:10.4310/MRL.2015.v22.n6.a5.
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