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Zariski Main Theorem
In algebraic geometry, Zariski's main theorem, proved by , is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of Zariski's connectedness theorem when the two varieties are birational. Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called Zariski's main theorem are as follows: *The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially Zariski's original form of his main theorem. *A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset. *The total transform of a normal point under a proper birational morphism is connected. *A closely related theorem of Grothendieck describes the structure of quasi-finite morphisms of schemes, which implies Zariski's origi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski's Connectedness Theorem
In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of varieties need not be birational. Zariski's connectedness theorem gives a rigorous version of the "principle of degeneration" introduced by Federigo Enriques, which says roughly that a limit of absolutely irreducible cycles is absolutely connected. Statement Suppose that ''f'' is a proper surjective morphism of varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular ... from ''X'' to ''Y'' such that the function field of ''Y'' is separably closed in that of ''X''. Then Zariski's connectedness theorem says that the inverse image of any norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-finite Morphism
In algebraic geometry, a branch of mathematics, a morphism ''f'' : ''X'' → ''Y'' of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions: * Every point ''x'' of ''X'' is isolated in its fiber ''f''−1(''f''(''x'')). In other words, every fiber is a discrete (hence finite) set. * For every point ''x'' of ''X'', the scheme is a finite κ(''f''(''x'')) scheme. (Here κ(''p'') is the residue field at a point ''p''.) * For every point ''x'' of ''X'', \mathcal_\otimes \kappa(f(x)) is finitely generated over \kappa(f(x)). Quasi-finite morphisms were originally defined by Alexander Grothendieck in SGA 1 and did not include the finite type hypothesis. This hypothesis was added to the definition in EGA II 6.2 because it makes it possible to give an algebraic characterization of quasi-finiteness in terms of stalks. For a general morphism and a point ''x'' in ''X'', ''f'' is said to be quasi-finite at ''x'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unibranch
In algebraic geometry, a local ring ''A'' is said to be unibranch if the reduced ring ''A''red (obtained by quotienting ''A'' by its nilradical) is an integral domain, and the integral closure ''B'' of ''A''red is also a local ring. A unibranch local ring is said to be geometrically unibranch if the residue field of ''B'' is a purely inseparable extension of the residue field of ''A''red. A complex variety ''X'' is called topologically unibranch at a point ''x'' if for all complements ''Y'' of closed algebraic subsets of ''X'' there is a fundamental system of neighborhoods (in the classical topology) of ''x'' whose intersection with ''Y'' is connected. In particular, a normal ring is unibranch. The notions of unibranch and geometrically unibranch points are used in some theorems in algebraic geometry. For example, there is the following result: Theorem Let ''X'' and ''Y'' be two integral locally noetherian schemes and f \colon X \to Y a proper dominant morphism. Denote their fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytically Normal
In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field. proved that if a local ring of an algebraic variety is normal, then it is analytically normal, which is in some sense a variation of Zariski's main theorem. gave an example of a normal Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ... local ring that is analytically reducible and therefore not analytically normal. References * * * * * Commutative algebra {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Immersion
Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YFriday album), 2001 * ''Open'' (Shaznay Lewis album), 2004 * ''Open'' (Jon Anderson EP), 2011 * ''Open'' (Stick Men album), 2012 * ''Open'' (The Necks album), 2013 * ''Open'', a 1967 album by Julie Driscoll, Brian Auger and the Trinity * ''Open'', a 1979 album by Steve Hillage * "Open" (Queensrÿche song) * "Open" (Mýa song) * "Open", the first song on The Cure album '' Wish'' Literature * ''Open'' (Mexican magazine), a lifestyle Mexican publication * ''Open'' (Indian magazine), an Indian weekly English language magazine featuring current affairs * ''OPEN'' (North Dakota magazine), an out-of-print magazine that was printed in the Fargo, North Dakota area of the U.S. * Open: An Autobiography, Andre Agassi's 2009 memoir Comput ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Morphism
In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is integral over k\left \right/math>. This definition can be extended to the quasi-projective varieties, such that a regular map f\colon X\to Y between quasiprojective varieties is finite if any point like y\in Y has an affine neighbourhood V such that U=f^(V) is affine and f\colon U\to V is a finite map (in view of the previous definition, because it is between affine varieties). Definition by Schemes A morphism ''f'': ''X'' → ''Y'' of schemes is a finite morphism if ''Y'' has an open cover by affine schemes :V_i = \mbox \; B_i such that for each ''i'', :f^(V_i) = U_i is an open affine subscheme Spec ''A''''i'', and the restriction of ''f'' to ''U''''i'', which induces a ring homomorphism :B_i \rightarrow A_i, makes ''A''''i'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-compact
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Morphism
In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism :\delta: X \to X \times_S X is a morphism determined by the universal property of the fiber product X \times_S X of ''p'' and ''p'' applied to the identity 1_X : X \to X and the identity 1_X. It is a special case of a graph morphism: given a morphism f: X \to Y over ''S'', the graph morphism of it is X \to X \times_S Y induced by f and the identity 1_X. The diagonal embedding is the graph morphism of 1_X. By definition, ''X'' is a separated scheme over ''S'' (p: X \to S is a separated morphism) if the diagonal morphism is a closed immersion. Also, a morphism p: X \to S locally of finite presentation is an unramified morphism if and only if the diagonal embedding is an open immersion. Explanation As an example, consider an algebraic variety over an algebraically closed field ''k'' and p: X \to \operatorname(k) the structure map. Then, identifying ''X'' with the set of its ''k''-rational points, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fulton–Hansen Connectedness Theorem
In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dimension at least 1. It is named after William Fulton and Johan Hansen, who proved it in 1979. The formal statement is that if ''V'' and ''W'' are irreducible algebraic subvarieties of a projective space ''P'', all over an algebraically closed field, and if : \dim(V) + \dim (W) > \dim (P) in terms of the dimension of an algebraic variety, then the intersection ''U'' of ''V'' and ''W'' is connected. More generally, the theorem states that if Z is a projective variety and f\colon Z \to P^n \times P^n is any morphism such that \dim f(Z) > n, then f^\Delta is connected, where \Delta is the diagonal in P^n \times P^n. The special case of intersections is recovered by taking Z = V \times W, with f the natural inclusion. See also * Zariski ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck's Connectedness Theorem
In mathematics, Grothendieck's connectedness theorem , states that if ''A'' is a complete Noetherian local ring whose spectrum is ''k''-connected and ''f'' is in the maximal ideal, then Spec(''A''/''fA'') is (''k'' − 1)-connected. Here a Noetherian scheme is called ''k''-connected if its dimension is greater than ''k'' and the complement of every closed subset of dimension less than ''k'' is connected. It is a local analogue of Bertini's theorem. See also * Zariski connectedness theorem *Fulton–Hansen connectedness theorem In mathematics, the Fulton–Hansen connectedness theorem is a result from intersection theory in algebraic geometry, for the case of subvarieties of projective space with codimension large enough to make the intersection have components of dim ... References Bibliography * * Theorems in algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
