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In recent years, considerable progress has been made in studying algebraic cycles using infinitesimal methods. These methods have usually been applied to Hodge-theoretic constructions such as the cycle class and the Abel-Jacobi map. Substantial advances have also occurred in the infinitesimal theory for subvarieties of a given smooth variety, centered around the normal bundle and the obstructions coming from the normal bundle's first cohomology group. Here, Mark Green and Phillip Griffiths set forth the initial stages of an infinitesimal theory for algebraic cycles. The book aims in part to understand the geometric basis and the limitations of Spencer Bloch's beautiful formula for the tangent space to Chow groups. Bloch's formula is motivated by algebraic K-theory and involves differentials over Q. The theory developed here is characterized by the appearance of arithmetic considerations even in the local infinitesimal theory of algebraic cycles. The map from the tangent space to the Hilbert scheme to the tangent space to algebraic cycles passes through a variant of an interesting construction in commutative algebra due to Angéniol and Lejeune-Jalabert. The link between the theory given here and Bloch's formula arises from an interpretation of the Cousin flasque resolution of differentials over Q as the tangent sequence to the Gersten resolution in algebraic K-theory. The case of 0-cycles on a surface is used for illustrative purposes to avoid undue technical complications.

512.73 --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Algebraic cycles. --- Hodge theory. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Cycles, Algebraic --- Algebraic cycles --- Hodge theory --- Addition. --- Algebraic K-theory. --- Algebraic character. --- Algebraic curve. --- Algebraic cycle. --- Algebraic function. --- Algebraic geometry. --- Algebraic number. --- Algebraic surface. --- Algebraic variety. --- Analytic function. --- Approximation. --- Arithmetic. --- Chow group. --- Codimension. --- Coefficient. --- Coherent sheaf cohomology. --- Coherent sheaf. --- Cohomology. --- Cokernel. --- Combination. --- Compass-and-straightedge construction. --- Complex geometry. --- Complex number. --- Computable function. --- Conjecture. --- Coordinate system. --- Coprime integers. --- Corollary. --- Cotangent bundle. --- Diagram (category theory). --- Differential equation. --- Differential form. --- Differential geometry of surfaces. --- Dimension (vector space). --- Dimension. --- Divisor. --- Duality (mathematics). --- Elliptic function. --- Embedding. --- Equation. --- Equivalence class. --- Equivalence relation. --- Exact sequence. --- Existence theorem. --- Existential quantification. --- Fermat's theorem. --- Formal proof. --- Fourier. --- Free group. --- Functional equation. --- Generic point. --- Geometry. --- Group homomorphism. --- Hereditary property. --- Hilbert scheme. --- Homomorphism. --- Injective function. --- Integer. --- Integral curve. --- K-group. --- K-theory. --- Linear combination. --- Mathematics. --- Moduli (physics). --- Moduli space. --- Multivector. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Open problem. --- Parameter. --- Polynomial ring. --- Principal part. --- Projective variety. --- Quantity. --- Rational function. --- Rational mapping. --- Reciprocity law. --- Regular map (graph theory). --- Residue theorem. --- Root of unity. --- Scientific notation. --- Sheaf (mathematics). --- Smoothness. --- Statistical significance. --- Subgroup. --- Summation. --- Tangent space. --- Tangent vector. --- Tangent. --- Terminology. --- Tetrahedron. --- Theorem. --- Transcendental function. --- Transcendental number. --- Uniqueness theorem. --- Vector field. --- Vector space. --- Zariski topology.

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The description for this book, Contributions to the Theory of Riemann Surfaces. (AM-30), Volume 30, will be forthcoming.

Riemann surfaces. --- Abelian integral. --- Algebraic curve. --- Algebraic function. --- Algebraic geometry. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Asymptotic formula. --- Automorphic function. --- Automorphism. --- Banach algebra. --- Bernhard Riemann. --- Boundary value problem. --- Bounded set (topological vector space). --- Coefficient. --- Compact Riemann surface. --- Compactification (mathematics). --- Complete metric space. --- Complex analysis. --- Complex manifold. --- Conformal map. --- Degeneracy (mathematics). --- Differential equation. --- Differential geometry. --- Differential of the first kind. --- Dimension (vector space). --- Dirichlet integral. --- Dirichlet problem. --- Dirichlet's principle. --- Divisor (algebraic geometry). --- Eigenvalues and eigenvectors. --- Elliptic function. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Existential quantification. --- Explicit formulae (L-function). --- Extremal length. --- Function (mathematics). --- Functional equation. --- Fundamental group. --- Fundamental theorem. --- Geometric function theory. --- Green's function. --- Harmonic conjugate. --- Harmonic function. --- Harmonic measure. --- Holomorphic function. --- Hyperbolic geometry. --- Hypergeometric function. --- Integral equation. --- Intersection (set theory). --- Interval (mathematics). --- Isometry. --- Isoperimetric inequality. --- Jordan curve theorem. --- Kähler manifold. --- Laplace's equation. --- Lebesgue integration. --- Linear differential equation. --- Linear map. --- Linear space (geometry). --- Mathematical physics. --- Mathematical theory. --- Mathematics. --- Meromorphic function. --- Metric space. --- Minkowski space. --- Operator (physics). --- Ordinary differential equation. --- Parametric equation. --- Parity (mathematics). --- Partial differential equation. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Quadratic differential. --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Riemann–Roch theorem. --- Ring (mathematics). --- Scalar (physics). --- Sign (mathematics). --- Simultaneous equations. --- Special case. --- Surjective function. --- Tensor density. --- Theorem. --- Theory of equations. --- Theory. --- Topology. --- Uniformization theorem. --- Uniformization. --- Uniqueness theorem. --- Variable (mathematics). --- Weierstrass theorem.

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Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.

Algebraic geometry --- Ordered algebraic structures --- Associative rings --- Abelian groups --- Functor theory --- Anneaux associatifs --- Groupes abéliens --- Foncteurs, Théorie des --- 512.73 --- 515.14 --- Functorial representation --- Algebra, Homological --- Categories (Mathematics) --- Functional analysis --- Transformations (Mathematics) --- Commutative groups --- Group theory --- Rings (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Algebraic topology --- Abelian groups. --- Associative rings. --- Functor theory. --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes abéliens --- Foncteurs, Théorie des --- Abelian group. --- Absolute value. --- Addition. --- Algebraic K-theory. --- Algebraic equation. --- Algebraic integer. --- Banach algebra. --- Basis (linear algebra). --- Big O notation. --- Circle group. --- Coefficient. --- Commutative property. --- Commutative ring. --- Commutator. --- Complex number. --- Computation. --- Congruence subgroup. --- Coprime integers. --- Cyclic group. --- Dedekind domain. --- Direct limit. --- Direct proof. --- Direct sum. --- Discrete valuation. --- Division algebra. --- Division ring. --- Elementary matrix. --- Elliptic function. --- Exact sequence. --- Existential quantification. --- Exterior algebra. --- Factorization. --- Finite group. --- Free abelian group. --- Function (mathematics). --- Fundamental group. --- Galois extension. --- Galois group. --- General linear group. --- Group extension. --- Hausdorff space. --- Homological algebra. --- Homomorphism. --- Homotopy. --- Ideal (ring theory). --- Ideal class group. --- Identity element. --- Identity matrix. --- Integral domain. --- Invertible matrix. --- Isomorphism class. --- K-theory. --- Kummer theory. --- Lattice (group). --- Left inverse. --- Local field. --- Local ring. --- Mathematics. --- Matsumoto's theorem. --- Maximal ideal. --- Meromorphic function. --- Monomial. --- Natural number. --- Noetherian. --- Normal subgroup. --- Number theory. --- Open set. --- Picard group. --- Polynomial. --- Prime element. --- Prime ideal. --- Projective module. --- Quadratic form. --- Quaternion. --- Quotient ring. --- Rational number. --- Real number. --- Right inverse. --- Ring of integers. --- Root of unity. --- Schur multiplier. --- Scientific notation. --- Simple algebra. --- Special case. --- Special linear group. --- Subgroup. --- Summation. --- Surjective function. --- Tensor product. --- Theorem. --- Topological K-theory. --- Topological group. --- Topological space. --- Topology. --- Torsion group. --- Variable (mathematics). --- Vector space. --- Wedderburn's theorem. --- Weierstrass function. --- Whitehead torsion. --- K-théorie