The subject of "K"-theory takes its name from a 1957 construction of Alexander Grothendieck which appeared in the Grothendieck–Riemann–Roch theorem, his generalization of Hirzebruch's theorem. Let "X" be a smooth algebraic variety. To each vector bundle on "X", Grothendieck associates an invariant, its "class". The set of all classes on "X" was called "K"("X") from the German "Klasse". By definition, "K"("X") is a quotient of the free abelian group on isomorphism classes of vector bundles on "X", and so it is an abelian group. If the basis element corresponding to a vector bundle "V" is denoted ["V"], then for each short exact sequence of vector bundles: