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FUNCTION SPACES AND CONTINUOUS ALGEBRAIC PAIRINGS FOR VARIETIES
"... Given a quasiprojective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of “smooth curves”) for studying t ..."
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Cited by 10 (6 self)
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Given a quasiprojective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of “smooth curves”) for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including: (1) the existence of cap and join product pairings in topological cycle theory, (2) the agreement of cup product and intersection product for topological cycle theory, (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product, and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.
THE HIGHER KTHEORY OF COMPLEX VARIETIES
, 2000
"... Abstract. Let X be a smooth complex variety, and let F be its function field. We prove that (after localizing at the prime 2) the Kgroups of F are divisible above the dimension of X, and that the Kgroups of X are divisiblebyfinite. We also describe the torsion in the Kgroups of F and X. In this ..."
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Cited by 2 (0 self)
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Abstract. Let X be a smooth complex variety, and let F be its function field. We prove that (after localizing at the prime 2) the Kgroups of F are divisible above the dimension of X, and that the Kgroups of X are divisiblebyfinite. We also describe the torsion in the Kgroups of F and X. In this paper we shall describe the abelian group structure of the algebraic Kgroups Kn of a complex variety X and its function field F, at least when n> dim(X) = tr. deg. (F) and Kn is localized at the prime ℓ = 2. Our results generalize Suslin’s description of the Ktheory of the complex numbers C in [Su1], and our earlier results for curves and surfaces in [PW]. Much of this paper depends upon the recent developments in motivic cohomology [SV,V2], related to the norm residue conjecture, that the the norm residue (F, Z/m) is an isomorphism for all m; see [BK, homomorphism KM n (F)/m → Hn et p. 118]. Here KM ∗ (F) is Milnor Ktheory and H ∗ et(F, Z/m) is étale cohomology. Since this conjecture might be settled soon, we have been encouraged to state
Application Of Motivic Complexes To Negligible Classes
, 1998
"... .  Lichtenbaum's complexes enable one to relate Galois cohomology to K cohomology groups. In this paper, we consider the first terms of the HochschildSerre spectral sequence for the cohomology of these complexes, which was developped by Kahn, in the case of quotients of "big" open sets in cellul ..."
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.  Lichtenbaum's complexes enable one to relate Galois cohomology to K cohomology groups. In this paper, we consider the first terms of the HochschildSerre spectral sequence for the cohomology of these complexes, which was developped by Kahn, in the case of quotients of "big" open sets in cellular varieties. In the particular case of a faithful representation W of a finite group G over an algebraically closed field k, this yields that the group of negligible classes in the cohomology group H 3 (G; Q=Z(2)) is canonically isomorphic to the second equivariant Chow group of a point. It also implies that the unramified classes in the cohomology group H 3 (k(W ) G ; (Q=Z) 0 (2)) come from the cohomology of G, which had been proved by Saltman when k is the field of complex numbers. Using the motivic complexes of Voevodsky, we then prove similar results in degrees four and five. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....