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The description for this book, K-Theory of Forms. (AM-98), Volume 98, will be forthcoming.

Category theory. Homological algebra --- 515.14 --- Algebraic topology --- 515.14 Algebraic topology --- Forms (Mathematics) --- K-theory --- Modules (Algebra) --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Homology theory --- Quantics --- Mathematics --- K-theory. --- Abelian group. --- Addition. --- Algebraic K-theory. --- Algebraic topology. --- Approximation. --- Arithmetic. --- Canonical map. --- Coefficient. --- Cokernel. --- Computation. --- Coprime integers. --- Coset. --- Direct limit. --- Direct product. --- Division ring. --- Elementary matrix. --- Exact sequence. --- Finite group. --- Finite ring. --- Free module. --- Functor. --- General linear group. --- Global field. --- Group homomorphism. --- Group ring. --- Homology (mathematics). --- Integer. --- Invertible matrix. --- Isomorphism class. --- Linear map. --- Local field. --- Matrix group. --- Maxima and minima. --- Mayer–Vietoris sequence. --- Module (mathematics). --- Monoid. --- Morphism. --- Natural transformation. --- Normal subgroup. --- P-group. --- Parameter. --- Power of two. --- Product category. --- Projective module. --- Quadratic form. --- Requirement. --- Ring of integers. --- Semisimple algebra. --- Sesquilinear form. --- Special case. --- Steinberg group (K-theory). --- Steinberg group. --- Subcategory. --- Subgroup. --- Subspace topology. --- Surjective function. --- Theorem. --- Theory. --- Topological group. --- Topological ring. --- Topology. --- Torsion subgroup. --- Triviality (mathematics). --- Unification (computer science). --- Unitary group. --- Witt group. --- K-théorie

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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

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This book contains accounts of talks held at a symposium in honorof John C. Moore in October 1983 at Princeton University, The workincludes papers in classical homotopy theory, homological algebra,rational homotopy theory, algebraic K-theory of spaces, and othersubjects.

Algebraic topology --- K-theory --- 512.73 --- 515.14 --- 512.73 Cohomology theory of algebraic varieties and schemes --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- Moore, John C. --- Abelian group. --- Adams spectral sequence. --- Adjoint functors. --- Adjunction (field theory). --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number field. --- Algebraic space. --- Algebraic topology. --- Algebraically closed field. --- Associative algebra. --- Boundary (topology). --- CW complex. --- Classification theorem. --- Closure (mathematics). --- Coalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Coproduct. --- Deformation theory. --- Degenerate bilinear form. --- Diagram (category theory). --- Differentiable manifold. --- Dimension (vector space). --- Division algebra. --- Eilenberg–Moore spectral sequence. --- Epimorphism. --- Exterior (topology). --- Formal power series. --- Free Lie algebra. --- Free algebra. --- Freudenthal suspension theorem. --- Function (mathematics). --- Function space. --- Functor. --- G-module. --- Galois extension. --- Global dimension. --- Group cohomology. --- Group homomorphism. --- H-space. --- Hilbert's Theorem 90. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Hopf invariant. --- Hurewicz theorem. --- Inclusion map. --- Inequality (mathematics). --- Integral domain. --- Isometry. --- Isomorphism class. --- K-theory. --- Lie algebra. --- Lie group. --- Limit (category theory). --- Loop space. --- Mathematician. --- Mathematics. --- Noetherian ring. --- Order topology. --- P-adic number. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal bundle. --- Principal ideal domain. --- Projective module. --- Projective plane. --- Pullback (category theory). --- Pushout (category theory). --- Ring of integers. --- Series (mathematics). --- Sheaf (mathematics). --- Simplicial category. --- Simplicial complex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Square (algebra). --- Stable homotopy theory. --- Steenrod algebra. --- Superalgebra. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Triviality (mathematics). --- Uniqueness theorem. --- Universal enveloping algebra. --- Vector bundle. --- Weak equivalence (homotopy theory). --- William Browder (mathematician). --- Géométrie algébrique --- K-théorie

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This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry. Of particular interest are the discussions concerning the Adams Conjecture, K-theories of finite fields, and Poincare duality. Because these applications have required repeated modifications of the original formulation of etale homotopy theory, the author provides a new treatment of the foundations which is more general and more precise than previous versions.One purpose of this book is to offer the basic techniques and results of etale homotopy theory to topologists and algebraic geometers who may then apply the theory in their own work. With a view to such future applications, the author has introduced a number of new constructions (function complexes, relative homology and cohomology, generalized cohomology) which have immediately proved applicable to algebraic K-theory.

Algebraic topology --- 512.73 --- 515.14 --- Homology theory --- Homotopy theory --- Schemes (Algebraic geometry) --- Geometry, Algebraic --- Deformations, Continuous --- Topology --- Cohomology theory --- Contrahomology theory --- Cohomology theory of algebraic varieties and schemes --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Homotopy theory. --- Homology theory. --- Abelian group. --- Adams operation. --- Adjoint functors. --- Alexander Grothendieck. --- Algebraic K-theory. --- Algebraic closure. --- Algebraic geometry. --- Algebraic group. --- Algebraic number theory. --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Algebraically closed field. --- Automorphism. --- Base change. --- Cap product. --- Cartesian product. --- Closed immersion. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex number. --- Complex vector bundle. --- Connected component (graph theory). --- Connected space. --- Coprime integers. --- Corollary. --- Covering space. --- Derived functor. --- Dimension (vector space). --- Disjoint union. --- Embedding. --- Existence theorem. --- Ext functor. --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Finite field. --- Finite group. --- Free group. --- Functor. --- Fundamental group. --- Galois cohomology. --- Galois extension. --- Geometry. --- Grothendieck topology. --- Homogeneous space. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy category. --- Homotopy group. --- Homotopy. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Inverse system. --- K-theory. --- Leray spectral sequence. --- Lie group. --- Local ring. --- Mapping cylinder. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Newton polynomial. --- Noetherian ring. --- Open set. --- Opposite category. --- Pointed set. --- Presheaf (category theory). --- Reductive group. --- Regular local ring. --- Relative homology. --- Residue field. --- Riemann surface. --- Root of unity. --- Serre spectral sequence. --- Shape theory (mathematics). --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of spectra. --- Simplex. --- Simplicial set. --- Special case. --- Spectral sequence. --- Surjective function. --- Theorem. --- Topological K-theory. --- Topological space. --- Topology. --- Tubular neighborhood. --- Vector bundle. --- Weak equivalence (homotopy theory). --- Weil conjectures. --- Weyl group. --- Witt vector. --- Zariski topology. --- Homologie --- Topologie algebrique --- Geometrie algebrique --- Homotopie