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

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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### References:

 [1] Baum, P; Fulton, W; MacPherson, R, Riemann-Roch for singular varieties, Publ. math. I.H.E.S., 45, 101-145, (1975) · Zbl 0332.14003 [2] Beilinson, A, Higher regulators and values of L-functions, Modern problems in mathematics, VINIT series, Vol. 24, 181-238, (1984), [Russian] [3] Beilinson, A, Height pairing between algebraic cycles, (1984), preprint · Zbl 0624.14005 [4] Beilinson, A, Letter to C. soulé, (November 1, 1982) [5] Bloch, S, Lectures on algebraic cycles, Duke university, math. series, No. IV, (1980) · Zbl 0436.14003 [6] Bloch, S, Algebraic K-theory and zeta functions of elliptic curves, (), 511-515 [7] Chevalley, C, Anneaux de Chow et applications, (), Secr. Math. Paris [8] Fulton, W, Intersection theory, () · Zbl 0541.14005 [9] {\scO. Gabber}, Preprint. [10] Gersten, S, Some exact sequences in the higher K-theory of rings, () · Zbl 0289.18011 [11] Gillet, H, Riemann Roch theorems for higher K-theory, Advan. in. math., 40, 203-289, (1981) · Zbl 0478.14010 [12] Grothendieck, Sga iv, () · Zbl 0197.47202 [13] Kratzer, C, Λ-structure en K-théorie algébrique, Comment. math. helv., 55, 233-254, (1970) · Zbl 0444.18008 [14] Landsburg, S, Relative cycles and algebraic K-theory, (1983), preprint [15] Levine, M, Cycles on singular varieties, (1983), preprint [16] Lichtenbaum, S, Values of zeta functions at non-negative integers, (1983), preprint · Zbl 0591.14014 [17] Quillen, D, Higher algebraic K-theory I, (), 85-147 · Zbl 0292.18004 [18] Roberts, J, Chow’s moving lemma, appendix to exposé of S. kleiman, () [19] {\scC. Soulé}, Opérations en K-théorie Algébrique, J. Canadian Math., in press. · Zbl 0575.14015 [20] Soulé, C, K-théorie et zéros aux points entiers de fonctions zêta, () · Zbl 0574.14010 [21] {\scA. A. Suslin}, On the K-theory of algebraically closed fields, preprint. · Zbl 0514.18008 [22] {\scA. A. Suslin}, On the K-theory of local fields, preprint. · Zbl 0548.12009 [23] Dayton, B; Weibel, C, A spectral sequence for the K-theory of affine glued schemes, (), 24-92 [24] {\scH. Gillet and Thomason}, The K-theory of strictly Hensel rings and a theorem of Suslin, preprint. · Zbl 0577.13009 [25] Kleiman, S, The transversality of a general translate, Compositio math., 38, 287-297, (1974) · Zbl 0288.14014
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