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Bloch's Higher Chow Groups
In algebraic geometry, Bloch's higher Chow groups, a generalization of Chow group, is a precursor and a basic example of motivic cohomology (for smooth varieties). It was introduced by Spencer Bloch and the basic theory has been developed by Bloch and Marc Levine. In more precise terms, a theorem of Voevodsky implies: for a smooth scheme ''X'' over a field and integers ''p'', ''q'', there is a natural isomorphism :\operatorname^p(X; \mathbb(q)) \simeq \operatorname^q(X, 2q - p) between motivic cohomology groups and higher Chow groups. Motivation One of the motivations for higher Chow groups comes from homotopy theory. In particular, if \alpha,\beta \in Z_*(X) are algebraic cycles in X which are rationally equivalent via a cycle \gamma \in Z_*(X\times \Delta^1), then \gamma can be thought of as a path between \alpha and \beta, and the higher Chow groups are meant to encode the information of higher homotopy coherence. For example,\text^*(X,0)can be thought of as the homotopy class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Group
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivic Cohomology
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology. Motivic homology and cohomology Let ''X'' be a scheme of finite type over a field ''k''. A key goal of algebraic geometry is to compute the Chow groups of ''X'', because they give strong information about all subvarieties of ''X''. The Chow groups of ''X'' have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for a closed subscheme ''Z'' of ''X'', there is an exact sequence of Chow groups, the localization sequence :CH_i(Z) \rightarrow CH_i(X) \rightarrow CH_i(X-Z) \rightarrow 0, whereas in topology this would be part of a long exact sequence. This problem was resolved by generalizing Chow groups to a bigrad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spencer Bloch
Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic ''K''-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Department of Mathematics of the University of Chicago. He is a member of the U.S. National Academy of Sciences and a Fellow of the American Academy of Arts and SciencesScholars, visiting faculty, leaders represent Chicago as AAAS fellows The University of Chicago Chronicle, April 30, 2009, Vol. 28 No. 15. Accessed January 12, 2010 and of the . At the |
Marc Levine (mathematician)
Marc N. Levine (born July 29, 1952 in Detroit, Michigan) is an American mathematician. Life and work Levine graduated from the Massachusetts Institute of Technology (bachelor's degree in 1974) and earned his doctorate in 1979 from Brandeis University under Teruhisa Matsusaka. He was assistant professor at the University of Pennsylvania in Philadelphia from 1979 and at Northeastern University (Boston), Northeastern University from 1984 in Boston, where he has been associate professor since 1986 and since 1988 professor. He was a visiting professor at University Duisburg-Essen, where he worked with Hélène Esnault. Since 2009 he has been Alexander von Humboldt Professor there. He was also a visiting scholar at Mathematical Sciences Research Institute, MSRI (1986, 1990), Max Planck Institute for Mathematics in Bonn (1983, 1987), Tata Institute of Fundamental Research (1988), at University of Washington, Caltech, University Paris VI and Henri Poincaré Institute. Levine works in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Scheme
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. Definition First, let ''X'' be an affine scheme of finite type over a field ''k''. Equivalently, ''X'' has a closed immersion into affine space ''An'' over ''k'' for some natural number ''n''. Then ''X'' is the closed subscheme defined by some equations ''g''1 = 0, ..., ''g''''r'' = 0, where each ''gi'' is in the polynomial ring ''k'' 'x''1,..., ''x''''n'' The affine scheme ''X'' is smooth of dimension ''m'' over ''k'' if ''X'' has dimension at least ''m'' in a neighborhood of each point, and the matrix of derivatives (∂''g''''i''/∂''x''''j'') has rank at least ''n''−''m'' everywhere on ''X''. (It follows that ''X'' has dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Cycle
In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space. While divisors on higher-dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ''W'' is given as \operatorname R for some ring ''R'' and ''X'', ''Y'' as \operatorname(R/I), \operatorname(R/J) for some ideals ''I'', ''J''. Thus, locally, the intersection X \cap Y is given as :\operatorname(R/(I+J)). Here, we used R/I \otimes_R R/J \simeq R/(I + J) (for this identity, see tensor product of modules#Examples.) Example: Let X \subset \mathbb^n be a projective variety with the homogeneous coordinate ring ''S/I'', where ''S'' is a polynomial ring. If H = \ \subset \mathbb^n is a hypersurface defined by some homogeneous polynomial ''f'' in ''S'', then : X \cap H = \operatorname(S/(I, f)). If ''f'' is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gysin Homomorphism
In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa. It was introduced by , and is generalized by the Serre spectral sequence. Definition Consider a fiber-oriented sphere bundle with total space ''E'', base space ''M'', fiber ''S''''k'' and projection map \pi: S^k \hookrightarrow E \stackrel M. Any such bundle defines a degree ''k'' + 1 cohomology class ''e'' called the Euler class of the bundle. De Rham cohomology Discussion of the sequence is clearest with de Rham cohomology. There cohomology classes are represented by differential forms, so that ''e'' can be represented by a (''k'' + 1)-form. The projection map \pi induces a map in cohomology H^\ast called its pullback \pi^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme-theoretic Intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes ''X'', ''Y'' of a scheme ''W'' is X \times_W Y, the fiber product of the closed immersions X \hookrightarrow W, Y \hookrightarrow W. It is denoted by X \cap Y. Locally, ''W'' is given as \operatorname R for some ring ''R'' and ''X'', ''Y'' as \operatorname(R/I), \operatorname(R/J) for some ideals ''I'', ''J''. Thus, locally, the intersection X \cap Y is given as :\operatorname(R/(I+J)). Here, we used R/I \otimes_R R/J \simeq R/(I + J) (for this identity, see tensor product of modules#Examples.) Example: Let X \subset \mathbb^n be a projective variety with the homogeneous coordinate ring ''S/I'', where ''S'' is a polynomial ring. If H = \ \subset \mathbb^n is a hypersurface defined by some homogeneous polynomial ''f'' in ''S'', then : X \cap H = \operatorname(S/(I, f)). If ''f'' is linear (deg = 1), it is called a hyperplane section. See also: Bertini's theorem. Now, a scheme-theoretic intersection may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dold–Kan Correspondence
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the nth homology group of a chain complex is the nth homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.) Example: Let ''C'' be a chain complex that has an abelian group ''A'' in degree ''n'' and zero in all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space K(A, n). There is also an ∞-category-version of the Dold–Kan correspondence. The book "Nonabelian Algebraic Topology" cited below has a Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a prev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
