Hochschild–Serre spectral sequence

Hochschild–Serre spectral sequence
Name	Hochschild–Serre spectral sequence
Field	Algebraic topology; Homological algebra; Group cohomology
Introduced	1953
Introduced by	Gerhard Hochschild; Jean-Pierre Serre

Hochschild–Serre spectral sequence is a fundamental computational tool in algebraic topology and homological algebra that relates the cohomology of a group extension to the cohomology of its quotient and subgroup. Originating in work by Gerhard Hochschild and Jean-Pierre Serre in the early 1950s, it systematically organizes higher extension data into a convergent spectral sequence, enabling computations in contexts connected to the Leray spectral sequence, Cartan–Leray spectral sequence, and techniques used by Henri Cartan, Jean Leray, and Edward Witten. Its influence extends to applications in the cohomology of Galois groups, Lie groups, and algebraic groups, and it interacts with constructions in the work of Alexander Grothendieck, Jean-Louis Koszul, and John Milnor.

Introduction

The Hochschild–Serre spectral sequence arises from an extension of groups 1 → N → G → Q → 1 and a G-module M, and organizes the relation among cohomology groups H^*(G,M), H^*(N,M), and H^*(Q,−). It plays a comparable role to the Lyndon–Hochschild–Serre spectral sequence in group cohomology literature and complements techniques developed by Samuel Eilenberg, Saunders Mac Lane, and Jean-Pierre Serre for group extensions, echoing constructions familiar from the Serre spectral sequence in the study of fiber bundles by Jean-Pierre Serre and Hatcher. The sequence is exact in a spectral sense and is frequently invoked alongside methods from Homological algebra, Category theory, and the theory of Derived functors.

Construction

Given a short exact sequence of discrete groups 1 → N → G → Q → 1 and a left G-module M, one constructs a first quadrant spectral sequence with E2-page E2^{p,q} = H^p(Q, H^q(N,M)) converging to H^{p+q}(G,M). The construction uses a double complex arising from bar resolutions due to Samuel Eilenberg and Saunders Mac Lane or from projective resolutions in the category of G-modules considered in Category theory settings studied by Alexander Grothendieck and Pierre Deligne. One forms a bicomplex by applying the Hom functor to a resolution of Z by free Z[G]-modules and then filters by columns or rows to obtain two filtrations as in the work of Jean Leray and Henri Cartan, producing the Hochschild–Serre E2-term via derived functors similar to Ext and Tor computations familiar from Emmy Noether’s era.

Convergence and Exact Sequences

The spectral sequence converges conditionally to the graded object associated to a natural filtration on H^{*}(G,M), mirroring convergence phenomena studied by Jean-Pierre Serre for the Serre spectral sequence and by Max Dehn in topological contexts. Under mild finiteness hypotheses (e.g., N or Q finite, or M of suitable cohomological dimension), the sequence collapses at a finite stage and yields five-term exact sequences analogous to the inflation–restriction sequence used by Emil Artin and Helmut Hasse in class field theory. These five-term exact sequences provide explicit connecting homomorphisms which are integral to computations in the cohomology of Galois groups in the tradition of Richard Dedekind and Emil Artin.

Examples and Computations

Classical computations include: - Cohomology of semidirect products G = N ⋊ Q where N and Q are cyclic groups, exemplified in calculations by Otto Schreier and George Mackey. - Galois cohomology computations for absolute Galois groups of number fields studied by Emil Artin, John Tate, and Alexander Grothendieck that underpin class field theory results like those of Helmut Hasse. - Lie algebra analogues where N is a normal Lie subgroup of a Lie group G, connecting to work of Nathan Jacobson, Claude Chevalley, and Élie Cartan. - Low-dimensional cases yielding explicit five-term sequences used by Jean-Pierre Serre in studying cohomological dimension and by John Milnor in investigations of algebraic K-theory.

Concrete computations often exploit known H^*(N,M) and H^*(Q,−) tables such as those tabulated by Israel Gelfand’s circle of collaborators or by later compendia inspired by Cartan and Eilenberg.

Applications

The Hochschild–Serre spectral sequence is applied in: - Galois cohomology and algebraic number theory following methods of John Tate and Emil Artin for class field theory and obstruction theory. - Classification of group extensions and group actions as in the program of Otto Schreier and I. Schur. - Study of principal bundles and fibration cohomology in the context of Jean-Pierre Serre’s topological work and in applications to Algebraic geometry by Alexander Grothendieck and Jean-Louis Verdier. - Computations in algebraic K-theory and motivic cohomology influenced by Daniel Quillen and Spencer Bloch. - Representation theory of finite groups and modular representation theory associated with names like Richard Brauer and Issai Schur.

Variants and Generalizations

Generalizations include spectral sequences for groupoids and stacks developed in the framework of Alexander Grothendieck’s topos theory, continuous cohomology for profinite groups as used in Galois theory by Jean-Pierre Serre and John Tate, and spectral sequences in the setting of Lie algebra cohomology tied to Cartan’s theory. Further categorical refinements relate to derived categories and triangulated categories studied by Bernard Keller and Amnon Neeman, and to equivariant cohomology theories influenced by Michael Atiyah and Raoul Bott.

Category:Spectral sequences