Poitou–Tate duality

E883485

duality theorem result in Galois cohomology

Poitou–Tate duality is a fundamental result in Galois cohomology that establishes deep duality relationships between global and local cohomology groups of number fields.

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Statements (47)

Predicate	Object
instanceOf	duality theorem ⓘ result in Galois cohomology ⓘ
appliesTo	absolute Galois groups of number fields ⓘ global fields ⓘ number fields ⓘ
assumes	finite Galois module with continuous action ⓘ
concerns	cohomology in degrees 0, 1, and 2 ⓘ cohomology of Galois groups of number fields ⓘ
context	cohomological dimension 2 of global Galois groups ⓘ
describes	compatibility of global and local pairings ⓘ
establishes	duality between global and local cohomology groups ⓘ
expressedAs	nine-term exact sequence in Galois cohomology ⓘ
field	Galois cohomology ⓘ algebraic number theory ⓘ
framework	cohomology of profinite groups ⓘ
generalizes	Tate local duality to global fields ⓘ
hasGeneralization	Poitou–Tate duality for p-adic representations NERFINISHED ⓘ Poitou–Tate duality in étale cohomology NERFINISHED ⓘ
implies	duality for Selmer and Tate–Shafarevich groups ⓘ finiteness properties of Galois cohomology groups ⓘ
involves	Pontryagin duality NERFINISHED ⓘ Tate–Shafarevich groups NERFINISHED ⓘ cohomology groups with restricted ramification ⓘ cohomology with compact support ⓘ discrete Galois modules ⓘ finite Galois modules ⓘ
namedAfter	Claude Poitou NERFINISHED ⓘ John Tate NERFINISHED ⓘ
provides	exact sequences relating global and local cohomology ⓘ orthogonality relations for local conditions ⓘ perfect pairings between cohomology groups ⓘ
relatedTo	Artin–Verdier duality NERFINISHED ⓘ Grothendieck duality NERFINISHED ⓘ local Tate duality ⓘ
relates	cohomology groups H^i(G_K,M) and H^{3-i}(G_K,M^∨(1)) ⓘ global Galois cohomology ⓘ local Galois cohomology ⓘ
requires	choice of a finite set of primes containing archimedean places ⓘ
typicalDomain	finite sets of places of a number field ⓘ
usedIn	Bloch–Kato conjectures NERFINISHED ⓘ Galois deformation theory NERFINISHED ⓘ Iwasawa theory NERFINISHED ⓘ arithmetic of elliptic curves ⓘ class field theory ⓘ modularity lifting theorems ⓘ study of Selmer groups ⓘ
uses	Tate local duality NERFINISHED ⓘ

Referenced by (1)

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Cohomologie Galoisienne → topic → Poitou–Tate duality ⓘ