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Algebraic Space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology. The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem). Definition There are two common ways to define algebraic spaces: they can be defined as either quotient ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack (mathematics)
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic, compatible geometrical objects (such as vector bundles on topological spaces) can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. It turns out that the general language for describing these coverings is that of a Grothendieck topology. Thus a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Manifold
In complex geometry, a Hopf manifold is obtained as a quotient of the complex vector space (with zero deleted) (^n\backslash 0) by a free action of the group \Gamma \cong of integers, with the generator \gamma of \Gamma acting by holomorphic contractions. Here, a ''holomorphic contraction'' is a map \gamma:\; ^n \to ^n such that a sufficiently big iteration \;\gamma^N maps any given compact subset of ^n onto an arbitrarily small neighbourhood of 0. Two-dimensional Hopf manifolds are called Hopf surfaces. Examples In a typical situation, \Gamma is generated by a linear contraction, usually a diagonal matrix q\cdot Id, with q\in a complex number, 0<, q, <1. Such manifold is called ''a classical Hopf manifold''. Properties A Hopf manifold isdiffeomorphic
In mathem ...
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Moishezon Manifold
In mathematics, a Moishezon manifold is a compact complex manifold such that the field of meromorphic functions on each component has transcendence degree equal the complex dimension of the component: :\dim_\mathbfM=a(M)=\operatorname_\mathbf\mathbf(M). Complex algebraic varieties have this property, but the converse is not true: Hironaka's example In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by . Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at mo ... gives a smooth 3-dimensional Moishezon manifold that is not an algebraic variety or scheme. showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric. showed that any Moishezon manifold carries an algebraic space structure; more precisely, the category of Moishezon spaces (similar to Moishezon manifolds, but are allowed to have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Space
An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also appear in other contexts. Definition Fix a field ''k'' with a valuation. Assume that the field is complete and not discrete with respect to this valuation. For example, this includes R and C with respect to their usual absolute values, as well as fields of Puiseux series with respect to their natural valuations. Let ''U'' be an open subset of ''k''''n'', and let ''f''1, ..., ''f''''k'' be a collection of analytic functions on ''U''. Denote by ''Z'' the common vanishing locus of ''f''1, ..., ''f''''k'', that is, let ''Z'' = . ''Z'' is an analytic variety. Suppose that the structure sheaf of ''U'' is \mathcal_U. Then ''Z'' has a structure sheaf \mathcal_Z = \mathcal_U / \mathcal_Z, where \mathcal_Z is the ideal generated by ''f'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Definition Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space. If ''W'' is a linear subspace of a finite-dimensional vector space ''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :\operatorname(W) = \dim(V) - \dim(W). It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the ambient space ''V:'' :\dim(W) + \operatorname(W) = \dim(V). S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hironaka's Example
In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by . Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at most 2 fail for smooth varieties of dimension at least 3. Hironaka's example Take two smooth curves ''C'' and ''D'' in a smooth projective 3-fold ''P'', intersecting in two points ''c'' and ''d'' that are nodes for the reducible curve C\cup D. For some applications these should be chosen so that there is a fixed-point-free automorphism exchanging the curves ''C'' and ''D'' and also exchanging the points ''c'' and ''d''. Hironaka's example ''V'' is obtained by gluing two quasi-projective varieties V_1 and V_2. Let V_1 be the variety obtained by blowing up P \setminus c along C and then along the strict transform of D, and let V_2 be the variety obtained by blowing up P\setminus d along ''D'' and then along the strict transform of ''C''. Si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-separated
In algebraic geometry, a morphism of schemes from to is called quasi-separated if the diagonal map from to is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi-compact). A scheme is called quasi-separated if the morphism to Spec is quasi-separated. Quasi-separated algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, ...s and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that is quasi-separated as part of the definition of an algebraic space or algebraic stack . Quasi-separated morphisms were introduced by as a generalization of separated morphisms. All separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated. Quasi-separa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indeterminate (variable)
In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objects such as polynomials and formal power series. In particular: * It does not designate a constant or a parameter of the problem. * It is not an unknown that could be solved for. * It is not a variable designating a function argument, or a variable being summed or integrated over. * It is not any type of bound variable. * It is just a symbol used in an entirely formal way. When used as placeholders, a common operation is to substitute mathematical expressions (of an appropriate type) for the indeterminates. By a common abuse of language, mathematical texts may not clearly distinguish indeterminates from ordinary variables. Polynomials A polynomial in an indeterminate X is an expression of the form a_0 + a_1X + a_2X^2 + \ldots + a_nX^n, where the ''a_i'' are called the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In abstract algebra, more specifically 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 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'' is any eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Function
In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are: * f(x) = 1/x * f(x) = \sqrt * f(x) = \frac Some algebraic functions, however, cannot be expressed by such finite expressions (this is the Abel–Ruffini theorem). This is the case, for example, for the Bring radical, which is the function implicitly defined by : f(x)^5+f(x)+x = 0. In more precise terms, an algebraic function of degree in one variable is a function y = f(x), that is continuous in its domain and satisfies a polynomial equation : a_n(x)y^n+a_(x)y^+\cdots+a_0(x)=0 where the coefficients are polynomial functions of , with integer coefficients. It can be shown that the same class of functions is obtained if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
