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.

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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.

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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.

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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.

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

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See also Swan's theorem.

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Extraordinary cohomology theory: REDIRECT Cohomology#Extraordinary cohomology theories...

Topological K-theory: In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michae...

Algebraic K-theory: In mathematics, algebraic K-theory is an advanced part of homological algebra concerned with defining and applying a sequence...

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