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Algebraic cycles and the Beilinson conjectures. (English) Zbl 0605.14017
Algebraic geometry, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, part I, Contemp. Math. 58, 65-79 (1986).
[For the entire collection see Zbl 0588.00015.]
This paper is effectively a synopsis of results to be found in a subsequent paper. Homotopy \((=\hom o\log y\)) groups related to the Chow groups are introduced. The basic properties of these homotopy groups, with in certain instances a modicum of proof, are described: functoriality, multiplicativity, relationship with algebraic K-theory, etc. - The author then relates:
(1) Bejlinson’s conjectures on the image of these homotopy groups in Deligne-Bejlinson cohomology under the regulator map;
(2) Lichtenbaum’s conjectures positing the existence of complexes in the derived category of étale sheaves on a scheme X related to the Milnor K-groups (together with related conjectures); and
(3) Soulé’s conjecture on the poles of the zeta function on a scheme X in terms of these homotopy groups.
Reviewer: P.Cherenack

14F35 Homotopy theory and fundamental groups in algebraic geometry
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14C25 Algebraic cycles
14F20 Étale and other Grothendieck topologies and (co)homologies
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)