The subject takes its name from a particular construction applied by Alexander Grothendieck in his proof of the RiemannRoch 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 Atiyah and 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 proof around 1962 of the Index Theorem.
In turn, JeanPierre Serre used the analogy of vector bundles[?] with projective modules[?] to found in 1959 what became algebraic Ktheory. He made Serre's conjecture, that projective modules over the ring of polynomials over a field are free modules[?]; this resisted proof for 20 years.
There followed a period in which there were various partial definitions of higher Kfunctors; until a comprehensive definition was given by Daniel Quillen[?] using homotopy theory[?].
The corresponding constructions involving an auxiliary quadratic form receive the general name Ltheory.
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