Algebraic cycles and higher K-theory. (English) Zbl 0608.14004

Bloch, Spencer

Adv. Math. 61, 267-304 (1986).

The main purpose of this paper is to lay the foundations of a theory of higher Chow groups, \(CH^*(X,n)\), \(n\geq 0\), where X is a quasi- projective scheme over a field k, in such a way as to generalize the Riemann-Roch theorem of Baum, Fulton and MacPherson and establish results which have been available for some time in higher algebraic K-theory. These Chow groups are defined as the homotopy groups of a simplicial complex of graded abelian groups associated to X, and this complex is conjectured to satisfy certain axioms of Beilinson and Lichtenbaum.
Among the properties established herein for \(CH^*(X,n)\) are: \((1)\quad functoriality\) (covariant for proper maps, contravariant for flat maps); \((2)\quad \hom otopy\); \((3)\quad localization\); \((4)\quad local\) to global spectral sequence; \((5)\quad multiplicative\) structure; and \((6)\quad Chern\) classes.

Reviewer: M.Stein

MSC:

14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
14C40	Riemann-Roch theorems
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14C05	Parametrization (Chow and Hilbert schemes)
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)

Keywords:

higher Chow groups; Riemann-Roch theorem

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