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


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)
Full Text: DOI


[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. 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, (Proc. ICM. Proc. ICM, Helsinki (1978)), 511-515
[7] Chevalley, C., Anneaux de Chow et applications, (“Sem. C. Chevalley,” \(2^è\) année (1958)), Secr. Math. Paris
[8] Fulton, W., Intersection Theory, (Ergebnisse Series (1984), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0541.14005
[10] Gersten, S., Some exact sequences in the higher \(K\)-theory of rings, (Algebraic \(K\)-Theory I. Algebraic \(K\)-Theory I, Springer Lecture Notes, No. 341 (1973), Springer-Verlag: Springer-Verlag Berlin) · 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, (Springer Lecture Notes, No. 225 (1971), Springer-Verlag: Springer-Verlag Berlin) · 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, (Lecture Notes in Math., No. 341 (1973), Springer-Verlag: Springer-Verlag Berlin), 85-147 · Zbl 0292.18004
[18] Roberts, J., Chow’s moving lemma, Appendix to exposé of S. Kleiman, (“Algebraic Geometry,” Oslo, 1970 (1972), Wolters-Noordhoff: Wolters-Noordhoff Groningen)
[20] Soulé, C., \(K\)-théorie et zéros aux points entiers de fonctions zêta, (Proc. ICM. Proc. ICM, Warszawa (1983)) · Zbl 0574.14010
[23] Dayton, B.; Weibel, C., A Spectral Sequence for the \(K\)-Theory of Affine Glued Schemes, (Springer Lecture Notes in Math., No. 854 (1980), Springer-Verlag: Springer-Verlag Berlin), 24-92
[25] Kleiman, S., The transversality of a general translate, Compositio Math., 38, 287-297 (1974) · Zbl 0288.14014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.