Cycles, Transfers, and Motivic Homology Theories. (AM-143)

Cycles, Transfers, and Motivic Homology Theories. (AM-143)

Vladimir Voevodsky
Andrei Suslin
Eric M. Friedlander
Copyright Date: 2000
Pages: 256
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    Cycles, Transfers, and Motivic Homology Theories. (AM-143)
    Book Description:

    The original goal that ultimately led to this volume was the construction of "motivic cohomology theory," whose existence was conjectured by A. Beilinson and S. Lichtenbaum. This is achieved in the book's fourth paper, using results of the other papers whose additional role is to contribute to our understanding of various properties of algebraic cycles. The material presented provides the foundations for the recent proof of the celebrated "Milnor Conjecture" by Vladimir Voevodsky.

    The theory of sheaves of relative cycles is developed in the first paper of this volume. The theory of presheaves with transfers and more specifically homotopy invariant presheaves with transfers is the main theme of the second paper. The Friedlander-Lawson moving lemma for families of algebraic cycles appears in the third paper in which a bivariant theory called bivariant cycle cohomology is constructed. The fifth and last paper in the volume gives a proof of the fact that bivariant cycle cohomology groups are canonically isomorphic (in appropriate cases) to Bloch's higher Chow groups, thereby providing a link between the authors' theory and Bloch's original approach to motivic (co-)homology.

    eISBN: 978-1-4008-3712-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-2)
  3. 1 Introduction
    (pp. 3-9)
    Eric M. Friedlander, A. Suslin and V. Voevodsky

    Our original goal which finally led to this volume was the construction of “motivic cohomology theory,” whose existence was conjectured by A.

    Beilinson and S. Lichtenbaum ([2], [3], [17], [18]). Even though this would seem to be achieved at the end of the third paper, our motivation evolved into a quest for a deeper understanding of various properties of algebraic cycles. Thus, several of the papers presented here do not deal directly with motivic cohomology but rather with basic questions about algebraic cycles.

    In this introduction, we shall begin with a short reminder of A. Beilinson’s formulation of motivic cohomology...

  4. 2 Relative Cycles and Chow Sheaves
    (pp. 10-86)
    Andrei Suslin and Vladimir Voevodsky

    LetXbe a scheme. A cycle onXis a formal linear combination of points of the Zariski topological space ofX.A cycle is called an effective cycle if all points appear in it with non negative coefficients. Suppose thatXis a projective scheme over a fieldkof characteristic zero. Then for any projective embedding$i:X \to {{\bf{P}}^n}$the classical construction produces a projective variety${C_r}(X,i)$called the Chow variety of effective cycles of dimensionronXsuch thatk-valued points of${C_r}(X,i)$are in natural bijection with effective cycles of dimensionronX.Moreover, for...

  5. 3 Cohomological Theory of Presheaves with Transfers
    (pp. 87-137)
    Vladimir Voevodsky

    Letkbe a field and$Sm/k$be the category of smooth schemes overk.In this paper we study contravariant functors from the category$Sm/k$to additive categories equipped withtransfer maps.More precisely we consider contravariant functors$F:{(Sm/k)^{op}} \to A$together with a family of morphisms${\phi _{{X {\left/ {} \right.} S}}}(Z):F(X) \to F(S)$given for any smooth curve$X \to S$

    over a smooth schemeSoverkand a relative divisorZonXoverSwhich is finite overS.If these maps satisfy some natural properties (see definition 3.1) such a collection of data is called apretheory over k.Some examples of pretheories are...

  6. 4 Bivariant Cycle Cohomology
    (pp. 138-187)
    Eric M. Friedlander and Vladimir Voevodsky

    The precursor of our bivariant cycle cohomology theory is the graded Chow group${A_*}(X)$of rational equivalence classes of algebraic cycles on a schemeXover a fieldk.In [1], S. Bloch introduced the higher Chow groups

    $CH*(X,*)$in order to extend to higher algebraic K-theory the relation established by A. Grothendieck between${K_0}(X)$and${A_*}(X)$. More recently, Lawson homology theory for complex algebraic varieties has been developed (cf. [13], [3]), in which the role of rational equivalence is replaced by algebraic equivalence, and a bivariant extension$L*H*(Y,X)$[7] has been introduced. This more topological approach suggested a duality...

  7. 5 Triangulated Categories of Motives Over a Field
    (pp. 188-238)
    Vladimir Voevodsky

    In this paper we construct for any perfect fieldka triangulated category$D{M^{eff}}(k)$which is called thetriangulated category of (effective) motivic complexescomplexesoverk(the minus sign indicates that we consider only complexes bounded from the above). This construction provides a natural categorical framework to study different algebraic cycle cohomology theories ([3],[13],[9],[7]) in the same way as the derived category of the etale sheaves provides a categorical framework for the etale cohomology. The first section of the paper may be considered as a long introduction. In §2.1 we give an elementary construction of a triangulated category$DM_{gm}^{eff}(k)$...

  8. 6 Higher Chow Groups and Etale Cohomology
    (pp. 239-254)
    Andrei A. Suslin

    The main purpose of the present paper is to relate the higher Chow groups of varieties over an algebraically closed field introduced by S. Bloch [B1] to etale cohomology. We follow the approach suggested by the author in 1987 during the Lumini conference on algebraic K-theory. The first 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 difficulty in the application of the results of [SV1] to higher Chow groups lies in the fact that a priori higher Chow are not defined as singular homology...