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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.

Bundeltheorie --- Cohomology [Sheaf ] --- Faisceaux [Théorie des ] --- Sheaf cohomology --- Sheaf theory --- Sheaves (Algebraic topology) --- Sheaves [Theory of ] --- Théorie des faisceaux --- Algebraic cycles --- Homology theory --- Algebraic cycles. --- Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Cycles, Algebraic --- Geometry, Algebraic --- Abelian category. --- Abelian group. --- Addition. --- Additive category. --- Adjoint functors. --- Affine space. --- Affine variety. --- Alexander Grothendieck. --- Algebraic K-theory. --- Algebraic cycle. --- Algebraically closed field. --- Andrei Suslin. --- Associative property. --- Base change. --- Category of abelian groups. --- Chain complex. --- Chow group. --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Cokernel. --- Commutative property. --- Commutative ring. --- Compactification (mathematics). --- Comparison theorem. --- Computation. --- Connected component (graph theory). --- Connected space. --- Corollary. --- Diagram (category theory). --- Dimension. --- Discrete valuation ring. --- Disjoint union. --- Divisor. --- Embedding. --- Endomorphism. --- Epimorphism. --- Exact sequence. --- Existential quantification. --- Field of fractions. --- Functor. --- Generic point. --- Geometry. --- Grothendieck topology. --- Homeomorphism. --- Homogeneous coordinates. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy. --- Injective sheaf. --- Irreducible component. --- K-theory. --- Mathematical induction. --- Mayer–Vietoris sequence. --- Milnor K-theory. --- Monoid. --- Monoidal category. --- Monomorphism. --- Morphism of schemes. --- Morphism. --- Motivic cohomology. --- Natural transformation. --- Nisnevich topology. --- Noetherian. --- Open set. --- Pairing. --- Perfect field. --- Permutation. --- Picard group. --- Presheaf (category theory). --- Projective space. --- Projective variety. --- Proper morphism. --- Quasi-projective variety. --- Residue field. --- Resolution of singularities. --- Scientific notation. --- Sheaf (mathematics). --- Simplicial complex. --- Simplicial set. --- Singular homology. --- Smooth scheme. --- Spectral sequence. --- Subcategory. --- Subgroup. --- Summation. --- Support (mathematics). --- Tensor product. --- Theorem. --- Topology. --- Triangulated category. --- Type theory. --- Universal coefficient theorem. --- Variable (mathematics). --- Vector bundle. --- Vladimir Voevodsky. --- Zariski topology. --- Zariski's main theorem. --- 512.73 --- 512.73 Cohomology theory of algebraic varieties and schemes --- Cohomology theory of algebraic varieties and schemes