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 formula_1 from the german "klasse". by definition, formula_1 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: