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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] |
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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...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] |
Morphism Of Finite Type
For a homomorphism ''A'' → ''B'' of commutative rings, ''B'' is called an ''A''-algebra of finite type if ''B'' is a 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'' 'x''1, ..., ''xn''is an ''A''-algebra of finite type, but it is not a finite ''A''-module unless ''A'' = 0 or ''n'' = 0. Another example of a finite-type morphism which is not finite is \mathbb \to \mathbb x,y]/(y^2 - x^3 - t). The analogous notion in terms of schemes is: a morphism ''f'': ''X'' → ''Y'' of schemes is of finite type if ''Y'' has a covering by affine open subschemes ''Vi'' = Spec ''Ai'' such that ''f''−1(''Vi'') has a finite covering by affine open subschemes ''Uij'' = Spec ''Bij'' with ''Bij'' an ''Ai''-algebra of finite type. One also says that ''X'' is of finite type over ''Y''. For exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Éléments De Géométrie Algébrique
The ''Éléments de géométrie algébrique'' ("Elements of algebraic geometry, Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French language, French, on algebraic geometry that was published (in eight parts or fascicle (book), fascicles) from 1960 through 1967 by the ''Institut des Hautes Études Scientifiques''. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of Scheme (mathematics), schemes, which he defined. The work is now considered the foundation stone and basic reference of modern algebraic geometry. Editions Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the ''Séminaire de géométrie algébrique'' (known as ''SGA''). Indeed, as explained by G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Scheme
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. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Scheme Theory
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Morphism
In algebraic geometry, a finite morphism between two affine varieties In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ... 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 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'': ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
