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| Hochschild–Serre | |
|---|---|
| Name | Hochschild–Serre |
| Field | Group cohomology; Homological algebra |
| Introduced | 1953 |
| Authors | Gerhard Hochschild; Jean-Pierre Serre |
| Notable for | Hochschild–Serre spectral sequence; analysis of group extensions; links to Galois cohomology |
Hochschild–Serre is principally associated with a spectral sequence and a collection of results in group cohomology established by Gerhard Hochschild and Jean-Pierre Serre in the mid‑20th century. The work provides tools for computing the cohomology of a group extension from the cohomology of a normal subgroup and the corresponding quotient, and it connects to topics studied by Emmy Noether, André Weil, Claude Chevalley, and later authors such as Serge Lang and Jean-Louis Koszul. The Hochschild–Serre apparatus became foundational for investigations involving Galois groups, Lie groups, algebraic groups, and structural questions addressed by Samuel Eilenberg, Saunders Mac Lane, and Henri Cartan.
The origin traces to collaboration between Gerhard Hochschild and Jean-Pierre Serre responding to problems in group cohomology and Galois cohomology influenced by earlier work of Emmy Noether, Richard Brauer, and Claude Chevalley. Published in the 1950s, their results built on the cohomological frameworks developed by Samuel Eilenberg and Saunders Mac Lane and on spectral sequence techniques used by Jean Leray and Jean-Louis Koszul. Influences include problems considered by André Weil in algebraic number theory and by Hyman Bass and John Tate in class field theory. The original formulation provided a bridge between computations in group extensions and structural analysis undertaken by Armand Borel and Harish-Chandra in the study of Lie groups and algebraic groups.
The Hochschild–Serre spectral sequence relates the cohomology of a group G with a normal subgroup N to the cohomology of N and the quotient Q = G/N. In formal terms it is a first quadrant spectral sequence E2^{p,q} = H^p(Q, H^q(N, M)) converging to H^{p+q}(G, M) for a G‑module M. The construction uses methods analogous to those in the spectral sequences of Leray, Cartan–Eilenberg, and Serre, and it exploits resolutions similar to those used by Eilenberg–Mac Lane and Henri Cartan. The Hochschild–Serre spectral sequence interacts with the Lyndon spectral sequence used in combinatorial group theory and complements techniques developed by Jean-Pierre Serre in his monograph on Galois cohomology and by John Tate in local duality. Variants appear in the work of Grothendieck on cohomology of sheaves and in constructions by Alexandre Grothendieck influencing Alexander Grothendieck’s school.
The sequence has been applied to compute cohomology groups of finite groups, profinite groups, and Lie groups. It is central in the analysis of inflation and restriction maps studied by Richard Brauer and John Tate and in explicit computations for groups considered by William Burnside and Otto Schreier. In Galois cohomology the Hochschild–Serre spectral sequence underlies descent arguments used by Jean-Pierre Serre, Ken Ribet, and Serge Lang in arithmetic investigations. It informs classification problems for group extensions pursued by Issai Schur and Reinhold Baer and plays a role in local and global duality theorems developed by Shafarevich and Tate. The tool has been used in structural results for algebraic groups by Borel and Tits and in deformation problems considered by Alexander Grothendieck and Pierre Deligne.
Generalizations include continuous and pro‑finite versions for profinite groups with coefficients in discrete modules, formulated in the context of Galois groups by John Tate and Serre. Derived functor formulations connect the Hochschild–Serre spectral sequence to the Grothendieck spectral sequence and to the Cartan–Eilenberg machinery used by Henri Cartan and Samuel Eilenberg. Equivariant and nonabelian extensions appear in work of Grothendieck and André Weil on nonabelian cohomology, and higher categorical analogues are studied in relation to derived algebraic geometry by authors influenced by Jacob Lurie and Bertrand Toën. Continuity and profinite adaptations are crucial in applications by Jean-Pierre Serre to Galois cohomology and in Iwasawa theory developed by Kenkichi Iwasawa and Ralph Greenberg.
Standard examples include the computation of H^*(G, M) for a semidirect product G = N ⋊ Q where N and Q are well‑understood, such as N = Z^n and Q = GL(n,Z) appearing in the work of Hermann Weyl and Emmy Noether. Classical computations for finite cyclic extensions relate to results of Ernst Artin and Emil Artin in class field theory, and explicit low‑dimensional calculations appear in papers by Schur and Hopf. The spectral sequence facilitates calculations for p‑groups analyzed by Philip Hall and Alperin, and for profinite Galois groups treated by Shafarevich and Iwasawa. Computational techniques borrow from resolutions used by Cartan–Eilenberg and from methods in the monograph of Brown on cohomology of groups.
Connections to algebraic geometry arise through the use of Hochschild–Serre methods in étale cohomology in the work of Alexander Grothendieck and Jean-Pierre Serre, and in descent theory as developed by Grothendieck and Alexandre Grothendieck’s school including Pierre Deligne and Michel Raynaud. In topology the sequence parallels spectral sequences used by Jean Leray and J. H. C. Whitehead and is related to fibrations studied by Serre in his work on homotopy groups. Interactions occur with the study of classifying spaces by G. W. Whitehead and with equivariant cohomology investigations by Atiyah and Bott, while applications to moduli problems appear in research by Deligne and M. Artin.