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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] |
Normal Ring
In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root of a monic polynomial with coefficients in ''A,'' then ''x'' is itself an element of ''A.'' Many well-studied domains are integrally closed: fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed. Note that integrally closed domains appear in the following chain of class inclusions: Basic properties Let ''A'' be an integrally closed domain with field of fractions ''K'' and let ''L'' be a field extension of ''K''. Then ''x''∈''L'' is integral over ''A'' if and only if it is algebraic over ''K'' and its minimal polynomial over ''K'' has coefficients in ''A''. In particular, this means that any element of ''L'' integral over ''A'' is root of a monic polynomial in ''A'' 'X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Closure
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a''''j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is a root of a monic polynomial over ''A''. The set of elements of ''B'' that are integral over ''A'' is called the integral closure of ''A'' in ''B''. It is a subring of ''B'' containing ''A''. If every element of ''B'' is integral over ''A'', then we say that ''B'' is integral over ''A'', or equivalently ''B'' is an integral extension of ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usual ... [...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] |
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 element of ''R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Ring
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring ''R'' form an ideal of ''R'', called the nilradical of ''R''; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring ''R/I'' is reduced if and only if ''I'' is a radical ideal. Let ''D'' be the set of all zero-divisors in a reduced ring ''R''. Then ''D'' is the union of all minimal prime ideals. Over a Noetherian ring ''R'', we say a finitely generated module ''M'' has locally constant rank if \mathfrak \mapsto \operatorname_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilradical Of A Ring
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: :\mathfrak_R = \lbrace f \in R \mid f^m=0 \text m\in\mathbb_\rbrace. In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways; see the article Radical of a ring for more on this. The nilradical of a Lie algebra is similarly defined for Lie algebras. Commutative rings The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent (by the binomial formula), and the product of any element with a nilpotent element is nilpotent (by commutativity). It can also be characterized as the intersection of all the prime ideals of the ring (in fact, it is the intersection of all minimal prime ideals). A ring is called reduced if it has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element ''a'' has the cancellation property, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entir ... [...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 ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a local ring and ''m'' is then its unique maximal ideal. This construction is 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 ''m''. Then the residue field is the quotient ring ''R''/''m''. Now suppose that ''X'' is a scheme and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some commutative ring. Considered in the neighbourhood ''U'', the point ''x'' correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Purely Inseparable Extension
In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x''''q'' = ''a'', with ''q'' a power of ''p'' and ''a'' in ''k''. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions. Purely inseparable extensions An algebraic extension E\supseteq F is a ''purely inseparable extension'' if and only if for every \alpha\in E\setminus F, the minimal polynomial of \alpha over ''F'' is ''not'' a separable polynomial.Isaacs, p. 298 If ''F'' is any field, the trivial extension F\supseteq F is purely inseparable; for the field ''F'' to possess a ''non-trivial'' purely inseparable extension, it must be imperfect as outlined in the above section. Several equivalent and more concrete definitions for the notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Morphism
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ''k'' is proper over ''k''. A scheme ''X'' of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space ''X''(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. Definition A morphism ''f'': ''X'' → ''Y'' of schemes is called universally closed if for every scheme ''Z'' with a morphism ''Z'' → ''Y'', the projection from the fiber product :X \times_Y Z \to Z is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ( GAII, 5.4.. One also says that ''X'' is proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominant Morphism
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 function. A regular map whose inverse is also regular is called biregular, and they are isomorphisms in the category of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on projective varieties – the weaker condition of a rational map and birational maps are frequently used as well. Definition If ''X'' and ''Y'' are closed subvarieties of \mathbb^n and \mathbb^m (so they are affine varieties), then a regular map f\colon X\to Y is the restriction of a polynomial map \mathbb^n\to \mathbb^m. Explicitly, it has the form: :f = (f_1, \dots, f_m) where the f_is are in the coordinate ring of ''X'': :k = k _1, \dots, x_nI, where ''I'' is the ideal defining ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birational
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. Birational maps Rational maps A rational map from one variety (understood to be irreducible) X to another variety Y, written as a dashed arrow , is defined as a morphism from a nonempty open subset U \subset X to Y. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset U is always dense in X, in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions. Birational maps A birational map from ''X'' to ''Y'' is a rational map such that there is a rational map inverse to ''f''. A birational map induces an isomorphism from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
