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
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,\mathbb {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,\mathbb {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} .}
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