Group cohomology
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| Group cohomology | |
|---|---|
| Name | Group cohomology |
| Field | Mathematics |
| Introduced | 1940s |
| Applications | Topology; Number theory; Algebraic geometry |
Group cohomology is the cohomological study of algebraic invariants associated to groups via modules, complexes, and derived functors. It connects algebraic topology, algebraic number theory, and algebraic geometry through calculations that classify extensions, detect obstructions, and measure deformation. Origins trace through interactions among algebraists and topologists in the twentieth century, influenced by work surrounding the Élie Cartan, Hassler Whitney, Shreeram Abhyankar schools and later developments by contributors such as Hermann Weyl, Claude Chevalley, Jean-Pierre Serre, Emil Artin, and John Tate.
Definition and basic constructions
Group cohomology is defined for a group G and a G-module M as right-derived functors of the fixed-point functor; this perspective uses resolutions or cochain complexes such as in the bar resolution, the inhomogeneous complex, or the standard complex. Early expositions relate to homological algebra frameworks developed by Samuel Eilenberg, Saunders Mac Lane, Henri Cartan, and Samuel Beckett (note: for mathematical context referencing historical figures), and to categorical language from Alexander Grothendieck and Jean-Louis Koszul. Concrete constructions use projective resolutions of the trivial module or injective resolutions in the category of modules over the group ring; these techniques employ concepts from Emmy Noether-inspired module theory and the David Hilbert syzygy theorem in algebraic examples.
Low-dimensional cohomology and interpretations
The cohomology groups H^0(G,M), H^1(G,M), and H^2(G,M) admit classical interpretations: H^0 describes invariants, H^1 classifies crossed homomorphisms and equivalence classes of splittings, while H^2 classifies equivalence classes of central extensions and obstruction classes. These interpretations were clarified in correspondence among Issai Schur, Otto Schreier, Emil Artin, Richard Brauer, Jean-Pierre Serre, and John Tate. Low-dimensional results connect to representation-theoretic phenomena in the work of Ferdinand Frobenius, William Burnside, and Issai Schur, and to obstruction theory studied by Armand Borel and Jean Leray.
Cohomological tools and computational techniques
Computational approaches exploit spectral sequences, long exact sequences, Shapiro's lemma, inflation-restriction sequences, and the Lyndon/Hochschild-Serre spectral sequence. Foundational contributors include Jean Leray, Günter Harder, Bertram Kostant, Hyman Bass, and Kenneth Brown. Methods draw on homological algebra formalism from Samuel Eilenberg, Saunders Mac Lane, and categorical techniques advanced by Alexander Grothendieck and Pierre Deligne. Computationally significant objects include projective resolutions, the bar complex, and continuous cohomology developed in contexts by John Tate and Claude Chevalley.
Relations with extensions, representations, and homological algebra
Cohomology classifies group extensions, measures obstructions to lifting projective representations to linear representations, and interacts with group actions on modules arising in representation theory of Niels Henrik Abel-inspired finite groups and in the modular representation theory of Richard Brauer and J. L. Alperin. Homological algebra tools such as Ext and Tor connect group cohomology with derived functors in the framework advanced by Jean-Pierre Serre, Alexander Grothendieck, Henri Cartan, and Samuel Eilenberg. Relations with group rings and module categories reflect studies by Emmy Noether, Richard Brauer, and Gerhard Hochschild.
Tate cohomology and periodicity
Tate cohomology refines ordinary cohomology for finite groups by combining homology and cohomology into a single graded theory defined by using complete resolutions; it exhibits periodicity phenomena for cyclic groups and connects to the work of John Tate, Irving Kaplansky, and Richard Swan. Important examples include periodic cohomology of cyclic groups and applications to class field theory influenced by Emil Artin and Helmut Hasse. Tate cohomology interacts with concepts from representation theory as developed by Issai Schur and Richard Brauer and feeds into modern duality theorems articulated in the work of Jean-Pierre Serre.
Applications in topology, number theory, and algebraic geometry
In algebraic topology, group cohomology interprets cohomology of classifying spaces and appears in the study of fibre bundles and characteristic classes linked to Frankel Hopf-style results and to work by René Thom and Hirzebruch. In number theory, Galois cohomology classifies algebraic extensions, Tate duality underlies local and global class field theory by John Tate, and connections to the arithmetic of elliptic curves were explored by André Weil, Goro Shimura, Yutaka Taniyama, and Bryan Birch. In algebraic geometry, group cohomology appears in the study of descent, obstructions to rational points, and étale cohomology frameworks advanced by Alexander Grothendieck, Jean-Pierre Serre, Alexander Grothendieck's school, and Pierre Deligne. Broader applications include deformation theory studied by Michael Artin, modularity questions influenced by Andrew Wiles, and reciprocity laws in the spirit of Kurt Hensel and Emil Artin.