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K-theory
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==Early history== The subject can be said to begin with [[Alexander Grothendieck]] (1957), who used it to formulate his [[Grothendieck–Riemann–Roch theorem]]. It takes its name from the German ''Klasse'', meaning "class".<ref>Karoubi, 2006</ref> Grothendieck needed to work with [[Coherent sheaf|coherent sheaves]] on an [[algebraic variety]] ''X''. Rather than working directly with the sheaves, he defined a group using [[isomorphism class]]es of sheaves as generators of the group, subject to a relation that identifies any extension of two sheaves with their sum. The resulting group is called ''K''(''X'') when only [[Locally free sheaf|locally free sheaves]] are used, or ''G''(''X'') when all are coherent sheaves. Either of these two constructions is referred to as the [[Grothendieck group]]; ''K''(''X'') has [[Cohomology|cohomological]] behavior and ''G''(''X'') has [[Homology (mathematics)|homological]] behavior. If ''X'' is a [[smooth variety]], the two groups are the same. If it is a smooth [[affine variety]], then all extensions of locally free sheaves split, so the group has an alternative definition. In [[topology]], by applying the same construction to [[vector bundle]]s, [[Michael Atiyah]] and [[Friedrich Hirzebruch]] defined ''K''(''X'') for a [[topological space]] ''X'' in 1959, and using the [[Bott periodicity theorem]] they made it the basis of an [[extraordinary cohomology theory]]. It played a major role in the second proof of the [[Atiyah–Singer index theorem]] (circa 1962). Furthermore, this approach led to a [[noncommutative topology|noncommutative]] K-theory for [[C*-algebra]]s. Already in 1955, [[Jean-Pierre Serre]] had used the analogy of [[vector bundle]]s with [[projective module]]s to formulate [[Quillen–Suslin theorem|Serre's conjecture]], which states that every finitely generated projective module over a [[polynomial ring]] is [[free module|free]]; this assertion is correct, but was not settled until 20 years later. ([[Swan's theorem]] is another aspect of this analogy.)
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