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| Tate–Nakayama duality | |
|---|---|
| Name | John Tate and Tadasi Nakayama |
| Field | Algebraic number theory, Cohomology |
| Notable works | Local class field theory, Galois cohomology |
Tate–Nakayama duality.
Tate–Nakayama duality is a reciprocity theorem in algebraic number theory and arithmetic geometry relating cohomology groups of finite Galois modules with Hom and Ext groups for Galois groups of local and global fields. It connects constructions from John Tate's work on Tate cohomology and local class field theory with Tadasi Nakayama's contributions to Galois cohomology and duality for continuous modules, providing a bridge between results used in the study of Weil group, Brauer group, Artin reciprocity, and the arithmetic of elliptic curves.
Tate–Nakayama duality refines classical local and global dualities by giving canonical isomorphisms between Tate cohomology groups and Pontryagin duals of cohomology groups for finite discrete modules over profinite Galois groups of fields such as Q_p, R (real numbers), and number fields like Q. The duality plays a central role alongside Tate duality, Poitou–Tate duality, and the Hochschild–Serre spectral sequence in analyzing arithmetic invariants that appear in the study of Selmer groups, Shafarevich–Tate groups, and the Brauer–Manin obstruction. Its formalism is used in proofs involving Artin–Verdier duality and comparisons with dualities for algebraic tori and abelian varieties.
Motivated by the development of class field theory by figures like Emil Artin, Helmut Hasse, and Teiji Takagi, Tate formulated duality theorems that synthesized earlier ideas from Richard Brauer's analysis of the Brauer group and Emil Noether's work on noncommutative extensions. Nakayama extended these frameworks in the context of cohomological methods associated to profinite groups studied by Jean-Pierre Serre and Shankar Sen. Subsequent influences include insights from Grothendieck's cohomological theories, the work of Alexander Grothendieck on duality in étale cohomology, and advances by Kazuya Kato and John Milnor in higher-dimensional generalizations.
Let G be the absolute Galois group of a local field such as Q_p or a global field like Q, and let M be a finite discrete G-module whose order is prime to the characteristic of the residue field; typical choices are torsion points of abelian varietys or finite multiplicative modules related to roots of unity. Tate–Nakayama duality asserts canonical isomorphisms between the Tate cohomology groups H^i(G,M) and the Pontryagin duals Hom(H^{2-i}(G,M^*),Q/Z) where M^* denotes the Cartier dual or local Tate dual of M. This formulation generalizes the local Tate duality for Galois modules and complements the global Poitou–Tate sequence used by researchers such as Jean-Louis Colliot-Thélène and Jürgen Neukirch.
The proof combines techniques from homological algebra, spectral sequences, and explicit computations in cyclic cohomology pioneered by Hermann Weyl's successors in representation theory. One constructs pairings via cup product combined with the fundamental class in H^2(G, \overline{K}^×) coming from local class field theory and employs the inflation-restriction sequence and the Hochschild–Serre spectral sequence to reduce to cyclic and procyclic subgroups such as decomposition groups at places studied by Weil and Chebotarev. Nakayama's lemma for cohomology and Tate's characterization of cohomological dimension for p-adic fields simplify vanishing results, while Pontryagin duality for locally compact abelian groups (used by analysts like L. Schwartz and L. Pontryagin) converts cohomological pairings into isomorphisms.
Standard examples include taking M = μ_n, the group of n-th roots of unity, where Tate–Nakayama duality recovers classical local statements about the Hilbert symbol and the pairing in the Brauer group of a local field such as Q_p or F_p((t)). For an elliptic curve E over a number field like Q with n-torsion E[n], the duality informs the relation between H^1(G,E[n]) and H^1(G,E^*[n]) and underlies explicit descent computations used by researchers such as John Cremona and Barry Mazur. Calculations for algebraic tori and norm one tori illustrate transfers and corestrictions that feature in work by Serre and Sansuc.
Tate–Nakayama duality is instrumental in proving finiteness and structural results about Shafarevich–Tate groups of abelian varietys, deriving exact sequences for Selmer groups, and analyzing obstructions like the Brauer–Manin obstruction to the Hasse principle studied by Colliot-Thélène and Jean-Jacques Sansuc. It feeds into explicit class field theory computations for cyclotomic extensions considered by Kummer and Leopoldt, and informs Iwasawa-theoretic analyses by figures such as Kenkichi Iwasawa and Ralph Greenberg. In the study of modular curves like X_0(N) and the arithmetic of modular forms relevant to Andrew Wiles's work, duality of cohomology groups underpins deformation theory arguments and control theorems used by Barry Mazur.
Generalizations include Poitou–Tate duality for global fields, Artin–Verdier duality in étale cohomology for arithmetic schemes like Spec Z and schemes over F_p, and Verdier duality in the derived category formulated by Jean-Louis Verdier and developed in the context of Grothendieck's six operations. Higher-dimensional analogues involve work of Kazuya Kato on higher local fields and Alexander Beilinson's conjectures connecting motivic cohomology to duality phenomena used by Spencer Bloch and Vladimir Voevodsky. Techniques from derived algebraic geometry and the study of Galois deformation rings by Mazur extend the conceptual reach of Tate–Nakayama duality into modern research on the Langlands program and arithmetic of motives.