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Chow's Lemma
Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: :If X is a scheme that is proper over a noetherian base S, then there exists a projective S-scheme X' and a surjective S-morphism f: X' \to X that induces an isomorphism f^(U) \simeq U for some dense open U\subseteq X. Proof The proof here is a standard one. Reduction to the case of X irreducible We can first reduce to the case where X is irreducible. To start, X is noetherian since it is of finite type over a noetherian base. Therefore it has finitely many irreducible components X_i, and we claim that for each X_i there is an irreducible proper S-scheme Y_i so that Y_i\to X has set-theoretic image X_i and is an isomorphism on the open dense subset X_i\setminus \cup_ X_j of X_i. To see this, define Y_i to be the scheme-theoretic image of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow's Theorem (other)
In mathematics, Chow's theorem may refer to a number of theorems due to Wei-Liang Chow: * Chow's theorem: The theorem that asserts that any analytic subvariety in projective space is actually algebraic. * Chow–Rashevskii theorem: In sub-Riemannian geometry, the theorem that asserts that any two points are connected by a horizontal curve. See also * Chow's lemma * Chow's moving lemma In algebraic geometry, Chow's moving lemma, proved by , states: given algebraic cycles ''Y'', ''Z'' on a nonsingular quasi-projective variety ''X'', there is another algebraic cycle ''Z' '' on ''X'' such that ''Z' '' is rationally equivalent to '' ... {{mathematical disambiguation Zhou, Weiliang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wei-Liang Chow
Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician and stamp collector born in Shanghai, known for his work in algebraic geometry. Biography Chow was a student in the US, graduating from the University of Chicago in 1931. In 1932 he attended the University of Göttingen, then transferred to the Leipzig University where he worked with van der Waerden. They produced a series of joint papers on intersection theory, introducing in particular the use of what are now generally called Chow coordinates (which were in some form familiar to Arthur Cayley). He married Margot Victor in 1936, and took a position at the National Central University in Nanjing. His mathematical work was seriously affected by the wartime situation in China. He taught at the National Tung-Chi University in Shanghai in the academic year 1946–47, and then went to the Institute for Advanced Study in Princeton, where he returned to his research. From 1948 to 197 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Morphism
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ''k'' is proper over ''k''. A scheme ''X'' of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space ''X''(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. Definition A morphism ''f'': ''X'' → ''Y'' of schemes is called universally closed if for every scheme ''Z'' with a morphism ''Z'' → ''Y'', the projection from the fiber product :X \times_Y Z \to Z is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ( GAII, 5.4.. One also says that ''X'' is proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Morphism
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Scheme
In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thus, a scheme is noetherian if and only if it is locally noetherian and quasi-compact. As with noetherian rings, the concept is named after Emmy Noether. It can be shown that, in a locally noetherian scheme, if \operatorname A is an open affine subset, then ''A'' is a noetherian ring. In particular, \operatorname A is a noetherian scheme if and only if ''A'' is a noetherian ring. Let ''X'' be a locally noetherian scheme. Then the local rings \mathcal_ are noetherian rings. A noetherian scheme is a noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-noetherian valuation ring. The definitions extend to formal schemes. Properties and Noetherian hypotheses Having a (locally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational Geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. Birational maps Rational maps A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow , is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions. Birational maps A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map inverse to ''f''. A birational map induces an isomorphism from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Algebraic Geometry
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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