the following variant of this is also used: instead of finitely generated projective (= locally free) modules, take finitely generated modules. the resulting "k"-groups are usually written "g"("r"). when "r" is a noetherian regular ring, then "g"- and "k"-theory coincide. indeed, the global dimension of regular rings is finite, i.e. any finitely generated module has a finite projective resolution "p" → "m", and a simple argument shows that the canonical map "k"(r) → "g"(r) is an isomorphism, with ["m"]=σ ± ["p"]. this isomorphism extends to the higher "k"-groups, too.