Tate duality

{{main|local Tate duality}}

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

:$\backslash displaystyle\; H^r(k,M)\backslash times\; H^\{2-r\}(k,M\text{'})\backslash rightarrow\; H^2(k,\backslash mathbb\{G\}\_m)=\backslash Q/\; \backslash Z$

where $M$ is a finite group scheme, $M\text{'}$ its dual $\backslash operatorname\{Hom\}(M,G\_m)$, and $\backslash mathbb\{G\}\_m$ is the multiplicative group.

For a local field of characteristic $p>0$, the statement is similar, except that the pairing takes values in $H^2(k,\; \backslash mu)\; =\; \backslash bigcup\_\{p\; \backslash nmid\; n\}\; \backslash tfrac\{1\}\{n\}\; \backslash Z/\backslash Z$.{{harvtxt|Neukirch|Schmidt|Wingberg|2000|loc=Theorem 7.2.6}} The statement also holds when $k$ is an Archimedean field, though the definition of the cohomology groups looks somewhat different in this case.

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

:$\backslash alpha\_\{r,\; M\}:\; H^r(k,\; M)\; \backslash rightarrow\; \{\backslash prod\_v\}\text{'}\; H^r(k\_v,\; M),$

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

:$\{\backslash prod\_v\}\text{'}\; H^r(k\_v,\; M)\; \backslash times\; \{\backslash prod\_v\}\text{'}\; H^\{2-\; r\}(k\_v,\; M\text{'})\; \backslash rightarrow\; \backslash Q/\backslash Z\; .$

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

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

:$\backslash text\{ker\}(\backslash alpha\_\{1,\; M\})\; \backslash times\; \backslash ker(\backslash alpha\_\{2,\; M\text{'}\})\; \backslash rightarrow\; \backslash Q/\backslash Z\; .$

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

:$0\; \backslash rightarrow\; H^0(k,\; M)\; \backslash rightarrow\; \{\backslash prod\_v\}\text{'}\; H^0(k\_v,\; M)\; \backslash rightarrow\; H^2(k,\; M\text{'})^*$

:$\backslash rightarrow\; H^1(k,\; M)\; \backslash rightarrow\; \{\backslash prod\_v\}\text{'}\; H^1(k\_v,\; M)\; \backslash rightarrow\; H^1(k,\; M\text{'})^*$

:$\backslash rightarrow\; H^2(k,\; M)\; \backslash rightarrow\; \{\backslash prod\_v\}\text{'}\; H^2(k\_v,\; M)\; \backslash rightarrow\; H^0(k,\; M\text{'})^*\; \backslash rightarrow\; 0.$

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 $S$ of $k$, with the above statements being the form of his theorems for the case where $S$ contains all places of $k$. For the more general result, see e.g.

{{harvtxt|Neukirch|Schmidt|Wingberg|2000|loc=Theorem 8.4.4}}.

Among other statements, Poitou–Tate duality establishes a perfect pairing between certain Shafarevich groups. Given a global field $k$, a set S of primes, and the maximal extension $k\_S$ which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of $\backslash operatorname\{Gal\}(k\_S/k)$ which vanish in the Galois cohomology of the local fields pertaining to the primes in S.See {{harvtxt|Neukirch|Schmidt|Wingberg|2000|loc=Theorem 8.6.8}} for a precise statement.

An extension to the case where the ring of S-integers $\backslash mathcal\{O\}\_S$ is replaced by a regular scheme of finite type over $\backslash operatorname\{Spec\}\; \backslash mathcal\{O\}\_S$ was shown by {{harvtxt|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.{{Cite journal |last=Česnavičius |first=Kęstutis |date=2015 |title=Poitou–Tate without restrictions on the order |url=https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2015/0022/0006/MRL-2015-0022-0006-a005.pdf |journal=Mathematical Research Letters |volume=22 |issue=6 |pages=1621–1666|doi=10.4310/MRL.2015.v22.n6.a5 }}

• Artin–Verdier duality
• Tate pairing

• {{Citation|arxiv=1709.06913|title=Poitou-Tate duality for arithmetic schemes|last1=Geisser|first1=Thomas H.|first2=Alexander|last2=Schmidt|journal=Compositio Mathematica|volume=154|issue=9|pages=2020–2044|year=2018|bibcode=2017arXiv170906913G|doi=10.1112/S0010437X18007340|s2cid=119735104 }}
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• {{Citation | last1=Poitou | first1=Georges | title=Cohomologie galoisienne des modules finis | url=https://books.google.com/books?id=E3GmAAAAIAAJ | publisher=Dunod | location=Paris | series=Séminaire de l'Institut de Mathématiques de Lille, sous la direction de G. Poitou. Travaux et Recherches Mathématiques |mr=0219591 | year=1967 | volume=13 | chapter=Propriétés globales des modules finis | pages=255–277}}
• {{Citation | last1=Tate | first1=John | author1-link=John Tate (mathematician) | title=Proceedings of the International Congress of Mathematicians (Stockholm, 1962) | chapter-url=http://mathunion.org/ICM/ICM1962.1/ | publisher=Inst. Mittag-Leffler | location=Djursholm | mr=0175892 | year=1962 | chapter=Duality theorems in Galois cohomology over number fields | pages=288–295 | url-status=dead | archiveurl=https://web.archive.org/web/20110717144510/http://mathunion.org/ICM/ICM1962.1/ | archivedate=2011-07-17 }}

Category:Algebraic number theory

Category:Galois theory

Category:Duality theories