K-theory

The topicK-theory spanssubjectsalgebraic topology, abstract algebrasome areasapplication like operator algebrasalgebraic geometry. It leads toconstructionfamiliesK-functors, which contain useful but often hard-to-compute information.

The subject takes its name fromparticular construction applied by Alexander Grothendieckhis proof ofRiemann-Roch theorem. In it,commutative monoidsheavesabelian groups under direct sum was converted intogroup, byformal additioninverses (an explicit wayexplainingleft adjoint). This construction was taken up by AtiyahHirzebruchdefine K(X) fortopological space X, by means onanalogous sum constructionvector bundles. This wasbasis offirst ofextraordinary cohomology theoriesalgebraic topology. It playedbig role inproof around 1962 ofIndex Theorem.

In turn, Jean-Pierre Serre usedanalogyvector bundlesprojective modulesfound1959 what became algebraic K-theory. He made Serre's conjecture, that projective modules overringpolynomials overfieldfree modules; this resisted proof20 years.

There followedperiodwhich there were various partial definitionshigher K-functors; untilcomprehensive definition was given by Daniel Quillen using homotopy theory.

The corresponding constructions involving an auxiliary quadratic form receivegeneral name L-theory.

See also Swan's theorem.