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Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.

Algebraic geometry --- Ordered algebraic structures --- Associative rings --- Abelian groups --- Functor theory --- Anneaux associatifs --- Groupes abéliens --- Foncteurs, Théorie des --- 512.73 --- 515.14 --- Functorial representation --- Algebra, Homological --- Categories (Mathematics) --- Functional analysis --- Transformations (Mathematics) --- Commutative groups --- Group theory --- Rings (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Algebraic topology --- Abelian groups. --- Associative rings. --- Functor theory. --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes abéliens --- Foncteurs, Théorie des --- Abelian group. --- Absolute value. --- Addition. --- Algebraic K-theory. --- Algebraic equation. --- Algebraic integer. --- Banach algebra. --- Basis (linear algebra). --- Big O notation. --- Circle group. --- Coefficient. --- Commutative property. --- Commutative ring. --- Commutator. --- Complex number. --- Computation. --- Congruence subgroup. --- Coprime integers. --- Cyclic group. --- Dedekind domain. --- Direct limit. --- Direct proof. --- Direct sum. --- Discrete valuation. --- Division algebra. --- Division ring. --- Elementary matrix. --- Elliptic function. --- Exact sequence. --- Existential quantification. --- Exterior algebra. --- Factorization. --- Finite group. --- Free abelian group. --- Function (mathematics). --- Fundamental group. --- Galois extension. --- Galois group. --- General linear group. --- Group extension. --- Hausdorff space. --- Homological algebra. --- Homomorphism. --- Homotopy. --- Ideal (ring theory). --- Ideal class group. --- Identity element. --- Identity matrix. --- Integral domain. --- Invertible matrix. --- Isomorphism class. --- K-theory. --- Kummer theory. --- Lattice (group). --- Left inverse. --- Local field. --- Local ring. --- Mathematics. --- Matsumoto's theorem. --- Maximal ideal. --- Meromorphic function. --- Monomial. --- Natural number. --- Noetherian. --- Normal subgroup. --- Number theory. --- Open set. --- Picard group. --- Polynomial. --- Prime element. --- Prime ideal. --- Projective module. --- Quadratic form. --- Quaternion. --- Quotient ring. --- Rational number. --- Real number. --- Right inverse. --- Ring of integers. --- Root of unity. --- Schur multiplier. --- Scientific notation. --- Simple algebra. --- Special case. --- Special linear group. --- Subgroup. --- Summation. --- Surjective function. --- Tensor product. --- Theorem. --- Topological K-theory. --- Topological group. --- Topological space. --- Topology. --- Torsion group. --- Variable (mathematics). --- Vector space. --- Wedderburn's theorem. --- Weierstrass function. --- Whitehead torsion. --- K-théorie