the group "k"("x") is now known as "k"("x"). upon replacing vector bundles by projective modules, "k" also became defined for non-commutative rings, where it had applications to group representations. atiyah and hirzebruch quickly transported grothendieck's construction to topology and used it to define topological k-theory. topological "k"-theory was one of the first examples of an extraordinary cohomology theory: it associates to each topological space "x" (satisfying some mild technical constraints) a sequence of groups "k"("x") which satisfy all the eilenberg–steenrod axioms except the normalization axiom. the setting of algebraic varieties, however, is much more rigid, and the flexible constructions used in topology were not available. while the group "k" seemed to satisfy the necessary properties to be the beginning of a cohomology theory of algebraic varieties and of non-commutative rings, there was no clear definition of the higher "k"("x"). even as such definitions were developed, technical issues surrounding restriction and gluing usually forced "k" to be defined only for rings, not for varieties.