Tate local duality
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| Tate local duality | |
|---|---|
| Name | Tate local duality |
| Field | Number theory, Algebraic geometry |
| Introduced | 1962 |
| Introduced by | John Tate |
| Related concepts | Local class field theory, Galois cohomology, Poitou–Tate duality |
Tate local duality is a fundamental duality theorem in Number theory and Algebraic geometry that relates Galois cohomology groups of local fields with coefficients in finite Galois modules to their Pontryagin duals. It provides a perfect pairing between cohomology groups in complementary degrees and underpins results in Local class field theory, the study of the Brauer group of local fields, and duality phenomena in arithmetic geometry such as the global Poitou–Tate duality and the study of Selmer groups for Elliptic curves.
Statement of the theorem
Let K be a non-archimedean local field such as a finite extension of Q_p or a field of formal Laurent series over a finite field, and let G_K denote its absolute Galois group related to Grothendieck's framework. For a finite discrete G_K-module M of order prime to the residue characteristic (or more generally a finite module), Tate local duality asserts a canonical perfect pairing between the continuous cohomology groups H^i(G_K,M) and H^{2-i}(G_K,M^*), where M^* = Hom(M, μ) is the Pontryagin dual built from the group of roots of unity μ in a separable closure of K. The pairing takes values in H^2(G_K,μ) which is canonically isomorphic to Q/Z via the local invariant map from Brauer group theory and Class field theory.
Background and definitions
Key objects and sources influencing the formulation include the absolute Galois group G_K studied by Emil Artin and Helmut Hasse in Local class field theory, the cohomological machinery developed by Jean-Pierre Serre and Alexander Grothendieck, and duality concepts appearing in the work of Claude Chevalley and Shafarevich.
Definitions: - Local field K: typical examples are finite extensions of Q_p (p-adic fields) or F_q((T)) related to André Weil's investigations. - Absolute Galois group G_K = Gal(K^sep/K) as in the tradition of Évariste Galois and later formalized by Emil Artin and John Tate. - Discrete G_K-module M: finite abelian groups with continuous G_K-action familiar from Kummer theory and the theory of Hilbert symbols. - Pontryagin dual M^* = Hom(M, μ_n) where μ_n denotes the group of n-th roots of unity studied by Kummer and Hilbert. - Cohomology H^i(G_K,−): continuous group cohomology developed in texts by Jean-Pierre Serre and applied in Tate cohomology contexts.
Important invariants used in the statement trace to the Brauer group computations of Richard Brauer and the invariant map of Albert Brauer and E. Artin that produces an isomorphism Br(K) ≅ Q/Z for local fields.
Proof outline
Tate's proof blends explicit cohomological computations, local class field theory, and duality of finite modules. Main steps: - Reduction to cyclic extensions inspired by methods of Noether and explicit cohomology computations for cyclic groups as treated by Emmy Noether and Kummer-theory expositions. - Construction of cup-product pairings H^i(G_K,M) × H^{2-i}(G_K,M^*) → H^2(G_K,μ) and identification of H^2(G_K,μ) with Q/Z using the local invariant map from Class field theory. - Use of Tate cohomology and the five-lemma in spectral sequences echoing techniques found in Cartan–Eilenberg homological algebra and Grothendieck's treatment of derived functors. - Verification of nondegeneracy via explicit dual bases in the cyclic case and passage to the general finite module by induction, paralleling approaches by Shafarevich and later expositions by Serre.
The proof is situated within the cohomological paradigm developed by John Tate and later refined in expositions by Milne and Neukirch.
Applications and consequences
Tate local duality is instrumental in: - Local reciprocity maps in Local class field theory and the classification of abelian extensions of local fields, building on work by Artin and Hasse. - Computations of the Brauer group of local and global fields, affecting statements in the Albert–Brauer–Hasse–Noether theorem. - Foundations of Poitou–Tate duality which relates global Galois cohomology of number fields studied by Poitou and generalized by Tate. - Analysis of Selmer and Sha groups for Elliptic curves and Abelian varietys as in papers by John Coates, Ralph Greenberg, and Mazur. - Duality theorems in étale cohomology framed by Grothendieck that feed into the Weil conjectures programme and theorems of Deligne. - Structural results in Iwasawa theory initiated by Kenkichi Iwasawa and developed by Barry Mazur and Ralph Greenberg.
Examples
- For K = finite extension of Q_p and M = μ_n, the pairing reduces to the classical Hilbert symbol studied by Hilbert and Hasse and computed via Kummer theory linked to Kummer and Artin reciprocity. - For unramified Galois modules arising from finite Étale cohomology of varieties over finite fields related to André Weil's zeta-function work, Tate local duality gives explicit dualities used by Grothendieck and Deligne. - For p-primary modules and K = Q_p, Tate local duality interfaces with Local Tate duality computations in Iwasawa theory as used by Iwasawa and Coates.
Variants and generalizations
Generalizations include: - Poitou–Tate duality for global fields combining local duality at all places as developed by Poitou and Tate. - Duality for p-adic representations and ((ϕ, Γ))-modules in p-adic Hodge theory advanced by Jean-Marc Fontaine and Kazuya Kato. - Duality in the setting of étale cohomology and derived categories explored by Grothendieck and Verdier leading to relative duality theorems used by Deligne and Illusie. - Noncommutative and higher-dimensional analogues appearing in arithmetic duality conjectures linked to work by Bloch, Kato, and researchers in motivic cohomology such as Voevodsky.