Tate cohomology

Tate cohomology
Name	Tate cohomology
Field	Algebraic topology; Number theory; Homological algebra
Introduced	1952
Introduced by	John Tate
Applications	Class field theory; Galois cohomology; Representation theory

Tate cohomology is a cohomology theory that combines Group cohomology and Group homology for finite Galois groups and finite groups into a single graded theory, with origins in the work of John Tate on Class field theory and Local fields. It plays a central role in the study of Galois module structure in number theory, connects to duality theorems such as Tate duality and the Poitou–Tate sequence, and interfaces with techniques from Homological algebra and derived categories developed by figures like Grothendieck, Cartan, and Eilenberg.

Definition and basic properties

Tate cohomology is defined for a finite group G and a G-module M, yielding groups denoted by \hat H^n(G,M) for integers n, combining positive-degree cohomology groups H^n(G,M) and negative-degree homology groups H_n(G,M) via a periodicity isomorphism for cyclic subgroups studied in work related to Herbrand and Noether. It satisfies long exact sequences analogous to those of Ext and Tor functors in Homological algebra as developed by Cartan and Eilenberg, and obeys functoriality with respect to restriction, corestriction, and inflation maps appearing in the theory of Galois extensions treated by Artin and Schreier. Fundamental properties include the vanishing of \hat H^n(G,M) for induced modules used in proofs by Tate and relationships to norm maps appearing in Class field theory and the Local reciprocity law.

Construction for finite groups

One common construction uses a complete resolution: splice a projective resolution of the trivial module as in the work of Cartan and Eilenberg with its dual to obtain an acyclic complex of projectives whose homology computes \hat H^n(G,M). This approach mirrors resolutions employed by Schur and Noether in representation theory and uses tools from Category theory introduced by Mac Lane. Alternatively, one may define \hat H^n(G,M) via stable module categories studied by Heller and Benson or via Tate's original definition using cohomology and homology groups together with the connecting homomorphism constructed by Tate in his papers influenced by correspondence with Artin and Hasse. The construction exploits transfer maps analogous to those used in the proof of the Herbrand–Ribet theorem and compatibility with induction and restriction functors prominent in the work of Mackey.

Relationship to group cohomology and homology

For n>0 there is a canonical isomorphism \hat H^n(G,M) \cong H^n(G,M), while for n<0 there is an identification \hat H^n(G,M) \cong H_{-n-1}(G,M) linking to classical homology groups studied by Eilenberg and Steenrod. At degree zero the group \hat H^0(G,M) measures coinvariants modulo norms, connecting with the Artin map in Class field theory and norm residue symbols appearing in the Brauer group context studied by Brauer and Albert. Tate cohomology satisfies spectral sequences such as the Lyndon/Hochschild–Serre spectral sequence developed by Lyndon and Hochschild for group extensions, and interacts with cup product structures explored by Milnor and Serre in relation to the Steinberg group and K-theory phenomena examined by Quillen.

Tate cohomology in algebraic number theory

Tate cohomology underpins duality results for Galois modules over global and local fields: Tate duality for local fields, the Poitou–Tate sequence for global fields, and reciprocity laws in Class field theory linking idele class groups to Galois groups as in work by Artin and Tate. It appears in the study of Selmer groups and Shafarevich–Tate groups for abelian varieties investigated by Tate, Shafarevich, and Mordell and in formulations of the Birch and Swinnerton-Dyer conjecture where Galois cohomology controls arithmetic invariants examined by Birch and Swinnerton-Dyer. In Iwasawa theory developed by Iwasawa and Mazur, Tate cohomology enters via Lambda-module structures and control theorems used by Kato and Greenberg in Euler system arguments, and it informs computations of the Tate module for elliptic curves and Abelian varietys considered by Weil.

Computations and examples

Classical computations include Tate cohomology for cyclic groups where periodicity of period two occurs, examples worked out by Herbrand and Lagrange-style theorems in the study of cyclotomic fields by Kummer and Leopoldt. Concrete calculations for finite groups such as Symmetric groups, Alternating groups, Dihedral groups, Quaternion groups, and p-groups connect to the modular representation theory of Brauer and Green, and to cohomology rings computed by Alperin, Benson, and Carlson. In arithmetic examples, Tate cohomology groups for the Galois action on units and ideal class groups feature in the proofs of Hilbert's theorem 90 and in explicit descriptions of the Hilbert class field as developed by Hilbert and Frobenius.

Generalizations and derived-category perspective

Generalizations extend Tate cohomology to profinite groups in Galois cohomology as in the work of Serre and Jannsen, to equivariant versions in Equivariant cohomology studied by Atiyah and Segal, and to stable module categories and triangulated categories influenced by Verdier and Neeman. In the derived-category language of Grothendieck and Deligne, Tate cohomology is realized via complete resolutions and total derived functors, linking to notions of local cohomology by Grothendieck and to compactly generated triangulated categories considered by Krause and Rickard. Recent developments connect Tate phenomena to Topological Hochschild homology and Algebraic K-theory as explored by Goodwillie and Weibel, and to higher categorical frameworks pursued by Lurie.

Category:Algebraic topology Category:Homological algebra Category:Number theory