|
Finite Morphism
In algebraic geometry, a finite morphism between two Affine variety, affine varieties X, Y is a dense Regular map (algebraic geometry), regular map which induces isomorphic inclusion k\left[Y\right]\hookrightarrow k\left[X\right] between their Coordinate ring, coordinate rings, such that k\left[X\right] is integral over k\left[Y\right]. This definition can be extended to the quasi-projective varieties, such that a Regular map (algebraic geometry), regular map f\colon X\to Y between quasiprojective varieties is finite if any point 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 scheme (mathematics), 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 restrictio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Closed Immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formalized by saying that f^\#:\mathcal_X\rightarrow f_\ast\mathcal_Z is surjective. An example is the inclusion map \operatorname(R/I) \to \operatorname(R) induced by the canonical map R \to R/I. Other characterizations The following are equivalent: #f: Z \to X is a closed immersion. #For every open affine U = \operatorname(R) \subset X, there exists an ideal I \subset R such that f^(U) = \operatorname(R/I) as schemes over ''U''. #There exists an open affine covering X = \bigcup U_j, U_j = \operatorname R_j and for each ''j'' there exists an ideal I_j \subset R_j such that f^(U_j) = \operatorname (R_j / I_j) as schemes over U_j. #There is a quasi-coherent sheaf of ideals \mathcal on ''X'' such that f_\ast\mathcal_Z\cong \mathcal_X/\mathcal an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Morphism Of Finite Type
In commutative algebra, given a homomorphism A\to B of commutative rings, B is called an A-algebra of finite type if B can be finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means that B is finitely generated as an A- module. For example, for any commutative ring A and natural number n, the polynomial ring A _1,\dots,x_n/math> is an A-algebra of finite type, but it is not a finite A-algebra unless A = 0 or n = 0. Another example of a finite-type homomorphism that is not finite is \mathbb \to \mathbb x,y]/(y^2 - x^3 - t). The analogous notion in terms of scheme (mathematics), schemes is that a morphism f:X\to Y of schemes is of finite type if Y has a covering by affine open subschemes V_i=\operatorname(A_i) such that f^(V_i) has a finite covering by affine open subschemes U_=\operatorname(B_) of X with B_ an A_i-algebra of finite type. One also says that X is of finite type over Y. For example, for any natural number n and field k, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Glossary Of Algebraic Geometry
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. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Noetherian Scheme
In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, where each A_i is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and compact. As with Noetherian rings, the concept is named after Emmy Noether. It can be shown that, in a locally Noetherian scheme, if \operatorname A is an open affine subset, then ''A'' is a Noetherian ring; in particular, \operatorname A is a Noetherian scheme if and only if ''A'' is a Noetherian ring. For a locally Noetherian scheme ''X,'' the local rings \mathcal_ are also Noetherian rings. A Noetherian scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring. The definitions extend to formal schemes. Properties and Noetherian hypotheses Having ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Glossary Of Algebraic Geometry
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called Grothendieck's relative point of view, "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des Hautes Études Scientifiques, Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He receive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1965, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Krull Dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' 'x''1, ..., ''x''''n''has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Wedderburn–Artin theorem ch ... [...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'' if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Going Up And Going Down
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions. The phrase going up refers to the case when a chain can be extended by "upward inclusion", while going down refers to the case when a chain can be extended by "downward inclusion". The major results are the Cohen–Seidenberg theorems, which were proved by Irvin S. Cohen and Abraham Seidenberg. These are known as the going-up and going-down theorems. Going up and going down Let ''A'' ⊆ ''B'' be an extension of commutative rings. The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in ''B'', each member of which lies over members of a longer chain of prime ideals in ''A'', to be able to be extended to the length of the chain of prime ideals in ''A''. Lying over and incomparability First, we fix some terminology. If \mathfrak and \mathfrak are prime ideals of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |