Algebraic cycles and higher K-theory

Here G,(X) is the Grothendieck group of coherent sheaves on X [13], gr; refers to the graded group defined by the y-filtration on G,(X) (cf. Kratzer [14], Soult [20]), and CH’(X) is the Chow group of codimension i algebraic cycles defined by Fulton [9]. The left-hand isomorphism is a formal consequence of the existence of a I-structure on G,(X) while the existence of r is the central theme of the B-F-M RR theorem. The main purpose of this paper is to define a theory of higher Chow groups CH*(X, n), n 2 0, so as to obtain isomorphisms