K-theory

In mathematics, K-theory is an extraordinary cohomology theory which consists of topological K-theory and algebraic K-theory and spans the subjects of algebraic topology, abstract algebra and some areas of application like operator algebras and algebraic geometry. It leads to the construction of families of K-functors, which contain useful but often hard-to-compute information.

Related Topics:
Mathematics - Extraordinary cohomology theory - Topological K-theory - Algebraic K-theory - Algebraic topology - Abstract algebra - Operator algebra - Algebraic geometry - Functor

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The subject takes its name from a particular construction applied by Alexander Grothendieck in his proof of the Grothendieck-Riemann-Roch theorem. In it, a commutative monoid of sheaves of abelian groups under direct sum was converted into a group, by the formal addition of inverses (an explicit way of explaining a left adjoint). This construction was taken up by Michael Atiyah and Friedrich Hirzebruch to define K(X) for a topological space X, by means on the analogous sum construction for vector bundles. This was the basis of the first of the extraordinary cohomology theories of algebraic topology. It played a big role in the second proof around 1962 of the Index Theorem. Furthermore this approach led to a noncommutative K-theory for C*-algebras.

Related Topics:
Alexander Grothendieck - Grothendieck-Riemann-Roch theorem - Monoid - Sheaves - Direct sum - Left adjoint - Michael Atiyah - Friedrich Hirzebruch - Topological space - Vector bundle - Algebraic topology - Index Theorem - Noncommutative - C*-algebra

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In turn, Jean-Pierre Serre used the analogy of vector bundles with projective modules to found in 1959 what became algebraic K-theory. He formulated Serre's conjecture, that projective modules over the ring of polynomials over a field are free modules; this resisted proof for 20 years.

Related Topics:
Jean-Pierre Serre - Vector bundle - Projective module - Algebraic K-theory - Serre's conjecture - Polynomial - Free module

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There followed a period in which there were various partial definitions of higher K-functors; until a comprehensive definition was given by Daniel Quillen using homotopy theory.

Related Topics:
Daniel Quillen - Homotopy theory

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The corresponding constructions involving an auxiliary quadratic form receive the general name L-theory.

Related Topics:
Quadratic form - L-theory

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See also Swan's theorem.

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