Merkurjev–Suslin theorem

Merkurjev–Suslin theorem
Name	Merkurjev–Suslin theorem
Field	Algebra, Number theory
Introduced	1982
Contributors	Alexander Merkurjev; Andrei Suslin

Merkurjev–Suslin theorem The Merkurjev–Suslin theorem is a foundational result in algebraic K-theory and quadratic form theory connecting the second Milnor K-group and the Brauer group for fields, with deep ties to Galois cohomology and central simple algebras. It establishes that the norm residue symbol induces an isomorphism between the n-torsion of the Brauer group and the degree-two Milnor K-group modulo n for fields containing sufficient roots of unity. This theorem shaped developments in the work of Jean-Pierre Serre, Alexander Grothendieck, Vladimir Voevodsky, and Pierre Deligne.

Statement of the theorem

The theorem asserts that for a field F containing a primitive n-th root of unity and for integer n invertible in F, the norm residue homomorphism from the Milnor K-group K_2^M(F)/n to the Galois cohomology group H^2(F, μ_n^{⊗2}) is an isomorphism. It identifies the n-torsion subgroup of the Brauer group Br(F)[n] with K_2^M(F)/n under the cup product pairing, relating central simple algebras classified by the Brauer group to symbols in Milnor K-theory. The statement played a pivotal role in subsequent conjectures by Jean-Pierre Serre and became a touchstone for work by Michel Demazure and Alexander Grothendieck on cohomological methods.

Historical context and motivation

Motivated by questions from Emmy Noether and the Brauer group studies of Richard Brauer, Emmy Noether, and Helmut Hasse, the problem of relating algebraic K-theory to division algebras drew attention from Ernst Witt and Tadeusz Kaczynski. Alexander Merkurjev and Andrei Suslin proved their theorem in 1982 after preceding contributions by Jacques Tits, Jean-Pierre Serre, and John Milnor on quadratic forms and Galois cohomology. The result anticipated later breakthroughs by Vladimir Voevodsky on the Bloch–Kato conjecture and built on concepts developed by Daniel Quillen and Henri Cartan.

Key concepts and prerequisites

Understanding the theorem requires familiarity with Milnor K-theory as introduced by John Milnor, the Brauer group studied by Richard Brauer and Helmut Hasse, and Galois cohomology developed by Jean-Pierre Serre and Claude Chevalley. One also needs the language of central simple algebras associated to Albert and Wedderburn, cyclic algebras of Nathan Jacobson, and the norm residue symbol related to Emil Artin and Helmut Hasse reciprocity laws. Foundational tools include étale cohomology from Alexander Grothendieck, the Merkurjev–Suslin norm residue homomorphism, and methods from Daniel Quillen’s algebraic K-theory framework.

Outline of the proof

Merkurjev and Suslin combined techniques from algebraic K-theory, cohomological invariants, and the theory of central simple algebras to show surjectivity and injectivity of the norm residue map. They used reduction to cyclic algebras building on work by Ernst Witt and Nathan Jacobson, analyzed cohomological dimension arguments influenced by Jean-Pierre Serre, and employed transfer arguments akin to those in Emil Artin’s class field theory. The proof exploited explicit symbol computations in Milnor K-theory, drew on the Merkurjev decomposition for algebras with involution related to Claude Chevalley’s methods, and anticipated later formalizations by Vladimir Voevodsky in motivic cohomology.

Consequences and applications

The theorem implies concrete identifications in the classification of central simple algebras studied by Richard Brauer and Albrecht Schröer, provides tools for computation in field invariants used by Helmut Hasse and Emil Artin, and underpins the Bloch–Kato conjecture resolved by Vladimir Voevodsky, Markus Rost, and others. It has applications in the study of quadratic forms traced to Ernst Witt and Alexander Grothendieck’s work on Brauer groups, influences on the study of division algebras examined by Nathan Jacobson, and connections to motivic cohomology developed by Spencer Bloch and Andrei Suslin. The theorem also informs explicit calculations in arithmetic geometry pursued by Jean-Pierre Serre and Pierre Deligne.

Examples and computations

For local and global fields considered by Helmut Hasse and Emil Artin, the Merkurjev–Suslin identification lets one compute Brauer group elements via Milnor K_2 symbols and cyclic algebras of Albert and Wedderburn type. Over number fields in the sense of Richard Dedekind and André Weil, the theorem yields explicit descriptions of n-torsion Brauer classes corresponding to Hilbert symbols studied by David Hilbert. For function fields over algebraically closed fields of interest to Alexander Grothendieck and Oscar Zariski, computations reduce to generators in Milnor K-theory related to Jacobson’s cyclic algebra constructions.

Category:Algebraic K-theory