Galois cohomology

Galois cohomology
Name	Galois cohomology
Field	Algebraic number theory, Algebraic geometry
Introduced	20th century
Notable	Évariste Galois, Claude Chevalley, Jean-Pierre Serre

Galois cohomology is a cohomological framework connecting the theory of Évariste Galois's field automorphisms with structural questions in Algebraic number theory, Algebraic geometry, and the arithmetic of elliptic curves. It organizes classical invariants such as Brauer group, Hilbert's Theorem 90, and norm-residue symbols into a graded sequence of groups equipped with functorial operations, enabling deep results in the work of Emil Artin, John Tate, Claude Chevalley, and Alexander Grothendieck.

Introduction

Galois cohomology arose from attempts by Évariste Galois and later Richard Dedekind to understand field extensions and was formalized through homological algebra by Jean-Pierre Serre and John Milnor, integrating ideas from Emil Artin's reciprocity, Alexander Grothendieck's schemes, and Jean-Louis Verdier's derived functors. It plays a central role in the theorems of Ernst Kummer and David Hilbert on norm residues, in André Weil's work on algebraic groups, and in Andrew Wiles's route to the Taniyama–Shimura conjecture through control of Selmer groups and Tate–Shafarevich groups.

Basic Definitions and Constructions

Given a profinite group Galois group Gal(E/F) coming from a field extension studied by Évariste Galois or a global field considered by Richard Dedekind, one defines continuous cohomology groups H^n(Gal(E/F), M) for discrete Gal-modules M, following formalism developed by Samuel Eilenberg and Saunders Mac Lane in the context of homological algebra and by Jean-Pierre Serre in number theory. The standard constructions use derived functors of invariants, injective resolutions à la Alexander Grothendieck and Henri Cartan, and explicit cocycle descriptions reminiscent of Niels Henrik Abel's reciprocity ideas. Functoriality ties these groups to restriction, inflation, corestriction, and connecting homomorphisms that mirror operations used by Emil Artin and John Tate.

Low-Degree Cohomology and Classical Applications

H^0 and H^1 recover fixed points and torsors, reflecting foundational work by Évariste Galois and David Hilbert; H^2 encodes obstructions such as elements of the Brauer group studied by Richard Brauer and used by Emil Artin in class field theory. Classical statements include Hilbert's Theorem 90 (vanishing of H^1 for multiplicative groups), Kummer theory connected to Ernst Kummer and Alexander Grothendieck's cohomological techniques, and the interpretation of central simple algebras via H^2 in the style of Richard Brauer and Max Albert. These low-degree computations underpin reciprocity laws developed by Emil Artin and reciprocity maps central to John Tate's cohomological treatment of global class field theory.

Cohomological Tools and Spectral Sequences

Advanced computations exploit spectral sequences from group extensions invoked by Jean Leray and later systematized in Jean-Pierre Serre's work, notably the Hochschild–Serre spectral sequence linking cohomology of intermediate fields to that of full Galois groups, paralleling techniques used by Henri Cartan and Samuel Eilenberg. Cup products and pairings introduced in homological algebra by Samuel Eilenberg and Saunders Mac Lane produce graded-commutative ring structures, while Tate cohomology and duality stems from contributions by John Tate and Alexander Grothendieck. Continuous cohomology methods for profinite groups rely on foundational work by Jean-Pierre Serre and John Tate, and derived categories introduced by Alexandre Grothendieck and Jean-Louis Verdier provide modern perspectives.

Galois Cohomology of Algebraic Groups and Torsors

One studies H^1 of algebraic groups to classify principal homogeneous spaces (torsors) following ideas of André Weil and applications by Armand Borel and Jean-Pierre Serre to reductive groups, where nonabelian cohomology features prominently in the work of Claude Chevalley and Alexander Grothendieck. The classification of forms of classical groups connects to the Kneser–Tits problem and results of Martin Kneser and Tonny Springer, while arithmetic applications to Shimura varieties and Langlands program contexts draw on cohomological descriptions used by Robert Langlands and Michael Harris.

Local and Global Duality Theorems

Duality theorems such as local Tate duality and Poitou–Tate duality were proved by John Tate and extend classical local reciprocity from Emil Artin and David Hilbert; they relate H^n of local fields with Pontryagin duals of H^{2-n}, and global duality intertwines local contributions via exact sequences reminiscent of formulations by Claude Chevalley and Jean-Pierre Serre. These theorems are essential in the study of Selmer groups in Iwasawa theory pioneered by Kenkichi Iwasawa and in the analysis of the Tate–Shafarevich group central to conjectures by André Weil and explored in the proof of the Birch–Swinnerton-Dyer conjecture contexts by Victor Kolyvagin and Karl Rubin.

Advanced Topics and Recent Developments

Recent progress weaves Galois cohomology into the Bloch–Kato conjecture resolved by contributions from Vladimir Voevodsky, Marc Levine, and Tommaso Geisser, linking Milnor K-theory of fields (developed by John Milnor) to Galois cohomology and norm-residue isomorphisms. Developments in the Langlands program by Robert Langlands, Michael Harris, and Laurent Lafforgue use cohomological techniques in the study of Galois representations appearing in the work of Pierre Deligne and Richard Taylor. Nonabelian generalizations, influenced by Alexander Grothendieck's anabelian geometry and advances by Shinichi Mochizuki, continue to drive research into field arithmetic and anabelian conjectures, while computational and explicit methods influenced by John Cremona and Nils Bruin apply Galois cohomology to concrete problems about rational points on curves studied by Andrew Wiles and Gerd Faltings.

Category:Algebraic number theory