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| Poitou–Tate sequence | |
|---|---|
| Name | Poitou–Tate sequence |
| Field | Algebraic number theory |
| Introduced | 1960s |
| Key concepts | Class field theory, Galois cohomology, Tate duality |
Poitou–Tate sequence The Poitou–Tate sequence is a fundamental long exact sequence in algebraic number theory linking global and local Galois cohomology groups via duality, combining ideas from class field theory, Tate duality, and global duality theorems. It sits at the intersection of work by Claude Chevalley, John Tate, Jean-Pierre Serre, and others, and plays a central role in modern arithmetic geometry, Iwasawa theory, and the study of Selmer groups and Shafarevich–Tate groups. The sequence relates Tate cohomology, local reciprocity laws from the Artin map, and global duality, with consequences for the structure of ideal class groups, Brauer groups, and arithmetic of abelian varieties over number fields.
The Poitou–Tate sequence arises when combining global duality theorems due to John Tate and local duality theorems due to Claude Chevalley and Jean-Pierre Serre with inputs from Emil Artin's reciprocity law and the Chebotarev density theorem. Its development was influenced by work of Alexander Grothendieck on duality in étale cohomology and by applications in the arithmetic of elliptic curves, abelian varieties, and Galois representations. The sequence is formulated for finite Galois modules over number fields such as Q or general number fields like Kreimer-type extensions, and is indispensable in the study of the Tate–Shafarevich group, the Selmer group, and the Brauer group.
Let F be a number field with ring of adèles related to places corresponding to embeddings into R and C, and let M be a finite discrete Galois module for the absolute Galois group Gal(Fbar/F) studied in the tradition of class field theory and Galois cohomology. The Poitou–Tate sequence gives a nine-term exact sequence relating the cohomology groups H^i(Gal(Fbar/F), M) and their local counterparts H^i(F_v, M) for places v of F, combined with the Pontryagin duals Hom(–, Q/Z) and Tate duals involving the Cartier dual M^*. The theorem synthesizes inputs from the Artin reciprocity law, the Global class field theory exact sequences, and duality results of Tate to produce exactness linking H^0, H^1, and H^2 groups globally and locally, culminating in a duality between the Shafarevich–Tate group and a global cohomology quotient analogous to statements in the work of John Milnor and Serre.
The formulation uses étale and Galois cohomology as developed by Jean-Pierre Serre in "Galois Cohomology", and rests on duality theorems by John Tate and foundational notions from Alexander Grothendieck's work on étale sites. Notation includes H^i(F, M) for Galois cohomology of the absolute Galois group Gal(Fbar/F), and H^i(F_v, M) for completions at places v linked to local fields such as Q_p and R. The Pontryagin dual M^∨ = Hom(M, Q/Z) and the Cartier dual M^* feature in exactness statements, as do the corestriction and restriction maps from the cohomology of finite extensions like those studied by Emile Artin and used in the proofs of the Brauer–Hasse–Noether theorem.
Proofs combine ingredients from local duality (Tate local duality for p-adic fields), global reciprocity (Artin global reciprocity), and spectral sequence arguments found in the work of Serre and Grothendieck. Key lemmas include the finiteness of H^i for finite modules over number fields (following methods of Chevalley and Shafarevich), the compatibility of cup products with local and global corestriction maps (as in Tate's duality papers), and the exactness of long sequences derived from mapping cones of localization maps, echoing techniques from Homological algebra authors such as Henri Cartan and Samuel Eilenberg. The proof constructs the long exact sequence by combining the global Poitou spectral sequence with local duality isomorphisms and invoking the Poitou–Tate duality framework developed in the literature of Mazur and Milne.
Variants extend the sequence to the context of profinite modules relevant to Iwasawa theory and to cohomology with compact support as developed by Milne and Grothendieck. Generalizations include analogues for function fields over finite fields influenced by Artin–Tate duality in the work of Tate and applications to Drinfeld modules, as well as derived-category formulations in the style of Pierre Deligne and Alexander Beilinson. There are also equivariant refinements involving Galois actions by groups like Gal(Qbar/Q) and non-commutative adaptations used in research by Coates, Fukaya, and Kato.
The sequence is used to analyze the structure of Selmer groups and the Tate–Shafarevich group for elliptic curves over number fields appearing in the conjectures of Birch and Swinnerton-Dyer. It underpins results about the finiteness of the Brauer group and the arithmetic duality theorems that influence work by Skorobogatov on descent and obstructions to the Hasse principle, and by Colliot-Thélène on rational points. In Iwasawa theory, the Poitou–Tate framework links λ- and μ-invariants studied by Kenkichi Iwasawa to duality phenomena, while in the theory of Galois representations it constrains deformation rings investigated by Mazur and Wiles. It also interfaces with the Tate conjecture and the formulation of Euler characteristic formulas in arithmetic homology theories, as developed by Bloch and Kato.
Computations typically involve finite cyclic modules like µ_n and Z/nZ over base fields such as Q, quadratic fields like Q(√-1), and cyclotomic extensions considered by Kummer and Kronecker. Explicit calculations recover the classical exact sequence relating the ideal class group, unit group, and the Brauer group as in the Brauer–Hasse–Noether theorem and show how Shafarevich–Tate groups for specific elliptic curves can be constrained via local conditions at primes like 2, 3, and p. Examples include Poitou–Tate computations for Kummer extensions relevant to Vandiver's conjecture and numerically explicit Selmer group analyses appearing in work by Cremona and Stein on modular elliptic curves.