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.

The semi-topological K-theory Ksemi ∗ (X) of a quasi-projective complex algebraic variety X is based on the notion of algebraic vector bundles modulo algebraic equivalence. This theory is given as the homotopy groups of an infinite loop space Ksemi (X) which is equipped with maps Kalg (X) → Ksemi (X), Ksemi (X) → Ktop(Xan) whose composition is the natural map from the algebraic K-theory of X to the topological K-theory of the underlying analytic space X an of X. We give an explicit description of Ksemi 0 (X) in terms of K0(X), a description of Ksemi q (−) in terms of Ksemi 0 (−) for projective varieties, a Poincaré duality theorem for projective varieties, and a computation of Ksemi (X) whenever X is a product of projective spaces or a smooth complete curve. For X a smooth quasi-projective variety, there are natural Chern class maps from K semi ∗ (X) to morphic cohomology compatible with similarly defined Chern class maps from algebraic K-theory to motivic cohomology and compatible with the classical Chern class maps from topological K-theory to the singular cohomology of Xan.

Notion of an oriented cohomology pretheory on algebraic varieties is introduced and a Riemann-Roch theorem for ring morphisms between oriented pretheories is proved. An explicit formula for the Todd genus related to a ring morphism is given.

Given a quasi-projective 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.

Abstract not found

In this paper, we re-formulate “morphic cohomology ” as introduced by the author and H.B. Lawson [F-L1] in such a way that it and “Lawson homology ” satisfy the list of basic properties codified by S. Bloch and A. Ogus [B-O]. This reformulation enables us to clarify and unify our previous definitions and provides this topological cycle theory with foundational properties which have proved useful for other cohomology theories. One formal consequence of these Bloch-Ogus properties is the existence of a local-to-global spectral sequence which should prove valuable for computations (as shown in Corollary 7.2). The basic result of this paper is that topological cycle cohomology theory (which agrees with morphic cohomology for smooth varieties) in conjunction with topological cycle homology theory (which is shown to always agree with Lawson homology) do indeed satisfy the Bloch-Ogus properties for a “Poincaré duality theory with supports ” on complex quasi-projective varieties. We view the challenge of verification of the Bloch-Ogus properties as worthy for several reasons. First, the properties require certain definitions and constructions

this paper we will reconstruct the above duality as a purely formal combination of a generalized form of Tate duality over p-adic fields and a form of Poincare duality for curves over arbitrary fields of characteristic zero. This gives a more conceptual proof of Lichtenbaum's result and an analogue in higher dimensions. Let j : X ! SpecK be a variety over a p-adic field, and consider the cohomological Brauer group Br(X) := H

2 Motivic cohomology and Bloch-Kato conjecture. 6 3 The approach to Bloch-Kato conjecture based on norm varieties. 14