Algebraic cycles and higher K-theory

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The relation between the category of coherent sheaves on an algebraic scheme X (i.e., a scheme of finite type over a field) and the group of algebraic cycles on X can be expressed in terms of the Riemann-Roth theorem of Baum, Fulton and McPherson (for simplicity we assume X equidimensional). 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 for-mal 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 0 grb G,(% & G,(X), P in 0 CH'O', n),,




Bloch, S. (1986). Algebraic cycles and higher K-theory. Advances in Mathematics, 61(3), 267–304.

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