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Noetherian Scheme
In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. 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 quasi-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. Let ''X'' be a locally noetherian scheme. Then the local rings \mathcal_ are 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 a (locally ... [...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] |
Abelian Varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chevalley's Theorem On Constructible Sets
In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure. They are used particularly in algebraic geometry and related fields. A key result known as ''Chevalley's theorem'' in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings (more specifically morphisms) of algebraic varieties (or more generally schemes). In addition, a large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible. Constructible sets also feature in the definition of various types of constructible sheaves in algebraic geometry and intersection cohomology. Definitions A simple definition, adequate in many situations, is that a constructible set is a finite union of locally closed sets. (A set is locally closed if it is the intersection of an open set and closed set.) However, a modification and another slightly weaker definition are needed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Ring
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic (algebra), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Covering
A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete space D and for every x \in X an Neighbourhood (mathematics), open neighborhood U \subset X, such that \pi^(U)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U is a homeomorphism for every d \in D . Often, the notion of a covering is used for the covering space E as well as for the map \pi : E \rightarrow X. The open sets V_ are called sheets, which are uniquely determined up to a homeomorphism if U is Connected space, connected. For each x \in X the discrete subset \pi^(x) is called the fiber of x. The degree of a covering is the cardinality of the space D. If E is Path connected, path-connected, then the covering \pi : E \rightarrow X is denoted as a path-connected covering. Examples * For every topological space X there exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. The philosopher Simone Weil was his sister. The writer Sylvie Weil is his daughter. Life André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71. Simone Weil, who would later become a famous philosopher, was Weil's younger sister and only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Muslim University in India. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he had taught himself Sanskrit in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Ring
In 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 ..., a topological ring is a Ring (algebra), ring R that is also a topological space such that both the addition and the multiplication are Continuity (topology), 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 (mathematics), 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#General topology, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Of Adeles
Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and literature * ''The Ring'' (franchise), a Japanese horror media franchise based on the novel series by Koji Suzuki ** ''Ring'' (novel series) *** ''Ring'' (Suzuki novel), 1991 ** ''Ring'' (film), or ''The Ring'', a 1998 Japanese horror film by Hideo Nakata *** ''The Ring'' (2002 film), an American horror film, remake of the 1998 Japanese film ** ''Ring'' (1995 film), a TV film ** ''Rings'' (2005 film), a short film by Jonathan Liebesman ** ''Rings'' (2017 film), an American horror film * ''Ring'' (Baxter novel), a 1994 science fiction novel * ''Ring'' (Alexis novel), a 2021 Canadian novel by André Alexis Gaming * ''Ring'' (video game), 1998 * Rings (''Sonic the Hedgehog''), a collectible in ''Sonic the Hedgehog'' games Music ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space \mathbf^3. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface :x^3+y^3+z^3+w^3=0 in \mathbf^3. Many properties of cubic surfaces hold more generally for del Pezzo surfaces. Rationality of cubic surfaces A central feature of smooth cubic surfaces ''X'' over an algebraically closed field is that they are all rational, as shown by Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by rational functions between the projective plane \mathbf^2 minus a lower-dimensional subset and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
