Abstract not found

Abstract not found

The purpose of this paper is to establish in Theorem 13.13 a spectral sequence from the motivic cohomology of a smooth variety X over a field F to the algebraic K-theory of X: E p,q 2 = Hp−q (X, Z(−q)) = CH −q (X, −p − q) ⇒ K−p−q(X). (13.13.1) Such a spectral sequence was conjectured by A. Beilinson [Be] as a natural analogue of the Atiyah-Hirzebruch spectral sequence from the singular cohomology to the topological K-theory of a topological space. The expectation of such a spectral sequence has provided much of the impetus for the development of motivic cohomology (e.g., [B1], [V2]) and should facilitate many computations in algebraic K-theory. In the special case in which X equals SpecF, this spectral sequence was established by S. Bloch and S. Lichtenbaum [B-L]. Our construction depends crucially upon the main result of [B-L], the existence of an exact couple relating the motivic cohomology of the field F to the multirelative K-theory of coherent sheaves on standard simplices over F (recalled as Theorem 5.5 below). A major step in generalizing the work of Bloch and Lichtenbaum is our reinterpretation of their spectral sequence in terms of the “topological filtration ” on the K-theory of the standard cosimplicial scheme ∆ • over F. We find that the spectral sequence arises from a tower of Ω-prespectra K( ∆ • ) = K 0 ( ∆ • ) ← − K 1 ( ∆ • ) ← − K 2 ( ∆ • ) ← − · · · Thus, even in the special case in which X equals SpecF, we obtain a much clearer understanding of the Bloch-Lichtenbaum spectral sequence which is essential for purposes of generalization. Following this reinterpretation, we proceed using techniques introduced by V. Voevodsky in his study of motivic cohomology. In order to do this, we provide an equivalent formulation of K-theory spectra associated to coherent sheaves on X with conditions on their supports K q ( ∆ • × X) which is functorial in X. We then Partially supported by the N.S.F. and the N.S.A.

this paper we study contravariant functors from the category Sm=k to additive 1

Abstract not found

Abstract not found

From the point of view taken in these lectures, motivic cohomology with coefficients in an abelian group A is a family of contravariant functors H p,q (−, A) : Sm/k → Ab from smooth schemes over a given field k to abelian groups, indexed by

Introduction The main purpose of the present paper is to relate the higher Chow groups of varieties over an algebraically closed eld introduced by S.Bloch [B1] to etale cohomology. We follow the approach suggested by the auther in 1987 during the Lumini conference on algebraic K-theory. The rst and most important step in this direction was done in [SV1], where singular cohomology of any qfh-sheaf were computed in terms of Ext-groups. The diculty in the application of the results of [SV1] to higher Chow groups lies in the fact that a priori higher Chow are not dened as singular homology of a sheaf. To overcome this diculty we prove that for an ane varietie X higher Chow groups CH i (X; n) of codimension i dimX may be computed using equidimensional cycles only(this is done in the rst two sections of the paper). In section 3 we generalize this result to all quasiprojective varieties over a eld of char

Let T be a triangulated category with coproducts, T c T the full subcategory of compact objects in T. If T is the homotopy category of spectra, Adams proved the following in [1]: All homological functors fT c g op ! Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other triangulated categories. In [36], it was proved that Adams' theorem remains true as long as T c is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in T. A puzzling open problem remained: Is every homological functor the restriction of a representable functor on T? In a recent paper, Beligiannis [5] made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories T = D(R) of rings, and homological functors fT c g op ! Ab which are not restrictions of representables. Contents

. The semi-topological K-theory of real varieties, KR semi (-), is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory, K semi (-), for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semi-topological equivalence), consideration of Z/2-equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic K-theory of real varieties. The authors verify that the semi-topological K-theory of a real variety X interpolates between the algebraic K-theory of X and Atiyah's Real K-theory of the associated Real space of complex points, X R (C). The resulting natural maps of spectra K alg (X) # KR semi (X) # KRtop (X R (C)) satisfy numerous good properties: the first map is a mod-n equivalence for any projective real variety and any n > 0; the second map is an equivalence for smooth projective curves...