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Functor Of Points
In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections, or one-to-one correspondence) the set of all morphisms S \to X. The functor ''F'' is then said to be naturally equivalent to the functor of points of ''X''; and the scheme ''X'' is said to '' represent'' the functor ''F'', and to ''classify'' geometric objects over ''S'' given by ''F''. A functor producing certain geometric objects over ''S'' might be represented by a scheme ''X''. For example, the functor taking ''S'' to the set of all line bundles over ''S'' (or more precisely ''n-''dimensional linear systems) is represented by the projective space X = \mathbb^. Another example is the Hilbert scheme ''X'' of a scheme ''Y'', which represents the functor sending a scheme ''S'' to the set of closed subschemes of Y\times S which are flat families over ''S''. In some app ... [...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] |
Berkovich Spectrum
In mathematics, a Berkovich space, introduced by , is a version of an analytic space over a non-Archimedean field (e.g. ''p''-adic field), refining Tate's notion of a rigid analytic space. Motivation In the complex case, algebraic geometry begins by defining the complex affine space to be \Complex^n. For each U\subset\Complex^n, we define \mathcal_U, the ring of analytic functions on U to be the ring of holomorphic functions, i.e. functions on U that can be written as a convergent power series in a neighborhood of each point. We then define a local model space for f_, \ldots, f_\in\mathcal_U to be :X:=\ with \mathcal_X=\mathcal_U/(f_, \ldots,f_). A complex analytic space is a locally ringed \Complex-space (Y, \mathcal_Y) which is locally isomorphic to a local model space. When k is a complete non-Archimedean field, we have that k is totally disconnected. In such a case, if we continue with the same definition as in the complex case, we wouldn't get a good analytic theory. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Field (scheme Theory)
The sheaf of rational functions ''KX'' of a scheme ''X'' is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of algebraic varieties, such a sheaf associates to each open set ''U'' the ring of all rational functions on that open set; in other words, ''KX''(''U'') is the set of fractions of regular functions on ''U''. Despite its name, ''KX'' does not always give a field for a general scheme ''X''. Simple cases In the simplest cases, the definition of ''KX'' is straightforward. If ''X'' is an ( irreducible) affine algebraic variety, and if ''U'' is an open subset of ''X'', then ''KX''(''U'') will be the fraction field of the ring of regular functions on ''U''. Because ''X'' is affine, the ring of regular functions on ''U'' will be a localization of the global sections of ''X'', and consequently ''KX'' will be the constant sheaf whose value is the fraction field of the global sections of ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal left ideal. * ''R'' has a unique maximal right ideal. * 1 ≠ 0 and the sum of any two non- units in ''R'' is a non-unit. * 1 ≠ 0 and if ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ring and \mathfrak is then its unique maximal ideal. In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the ''natural domain'' for the coordinates of the point. Definition Suppose that R is a commutative local ring, with maximal ideal \mathfrak. Then the residue field is the quotient ring R/\mathfrak. Now suppose that X is a scheme and x is a point of X. By the definition of a scheme, we may find an affine neighbourhood \mathcal = \text(A) of x, with some commutative ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Site
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda Lemma
In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It also generalizes the information-preserving relation between a term and its continuation-passing style transformation from programming language theory. It allows the embedding of any locally small category into a category of functors ( contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. Generalities The Yoneda lemma suggests that instead of studyi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Ring
In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive topological group and a multiplicative topological semigroup. Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field. General comments The group of units R^\times of a topological ring R is a topological group when endowed with the topology coming from the embedding of R^\times into the product R \times R as \left(x, x^\right). However, if the unit group is endowed with the subspace topology as a subspace of R, it may not be a topological group, because inversion on R^\times need not be continuous with respect to the subspace topology. An example of this situation is the adele ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yoneda's Lemma
In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It also generalizes the information-preserving relation between a term and its Continuation-passing style, continuation-passing style transformation from programming language theory. It allows the Subcategory#Embeddings, embedding of any locally small category into a category of functors (Functor#Covariance and contravariance, contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basepoint-free
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective spaces. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bundle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear System Of Divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a ''linear system'' of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisor (algebraic geometry), divisors ''D'' on a general Scheme (mathematics), scheme or even a ringed space (X, \mathcal_X). Linear systems of dimension 1, 2, or 3 are called a Pencil (mathematics), pencil, a net, or a web, respectively. A map determined by a linear system is sometimes called the Kodaira map. Definitions Given a general variety X, two divisors D,E \in \text(X) are linearly equivalent if :E = D + (f)\ for some non-zero rational function f on X, or in other words a non-zero element f of the Function field of an algebraic variety, func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
