"k"-theory now appeared to be a homology theory for rings and a cohomology theory for varieties. however, many of its basic theorems carried the hypothesis that the ring or variety in question was regular. one of the basic expected relations was a long exact sequence (called the "localization sequence") relating the "k"-theory of a variety "x" and an open subset "u". quillen was unable to prove the existence of the localization sequence in full generality. he was, however, able to prove its existence for a related theory called "g"-theory (or sometimes "k"′-theory). "g"-theory had been defined early in the development of the subject by grothendieck. grothendieck defined "g"("x") for a variety "x" to be the free abelian group on isomorphism classes of coherent sheaves on "x", modulo relations coming from exact sequences of coherent sheaves. in the categorical framework adopted by later authors, the "k"-theory of a variety is the "k"-theory of its category of vector bundles, while its "g"-theory is the "k"-theory of its category of coherent sheaves. not only could quillen prove the existence of a localization exact sequence for "g"-theory, he could prove that for a regular ring or variety, "k"-theory equaled "g"-theory, and therefore "k"-theory of regular varieties had a localization exact sequence. since this sequence was fundamental to many of the facts in the subject, regularity hypotheses pervaded early work on higher "k"-theory.