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Nagata's Compactification Theorem
In algebraic geometry, Nagata's compactification theorem, introduced by , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme ''S'' can be factored into an open immersion followed by a proper morphism. Nagata's original proof used the older terminology of Zariski–Riemann spaces and valuation theory, which sometimes made it hard to follow. Deligne showed, in unpublished notes expounded by Conrad, that Nagata's proof can be translated into scheme theory and that the condition that ''S'' is Noetherian can be replaced by the much weaker condition that ''S'' is quasi-compact and quasi-separated. gave another scheme-theoretic proof of Nagata's theorem. An important application of Nagata's theorem is in defining the analogue in algebraic geometry of cohomology with compact support, or more generally higher direct image functors with proper support. The idea is that given ... [...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 topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Image Functor
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor from the category of sheaves on ''Y'' to the category of sheaves on ''X''. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features. Definition Suppose we are given a sheaf \mathcal on Y and that we want to transport \mathcal to X using a continuous map f\colon X\to Y. We will call the result the ''inverse image'' or pullback sheaf f^\mathcal. If we try to imitate the direct image by setting :f^\mathcal(U) = \mathcal(f(U)) for each open set U of X, we immediately run into a problem: f(U) is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X- modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SGA 4
SGA may refer to: * Old Irish language (ISO 639-3 code) * ''Schwarz-Gelbe Allianz'' (Black-Yellow Alliance), an Austrian political party * Second-generation antipsychotics * Séminaire de Géométrie Algébrique du Bois Marie, an influential mathematical seminar and the series of books it produced. * SGA Airlines (Siam General Aviation), a regional airline in Thailand * SG&A, Selling, General and Administrative expenses in income statements * Simple Genetic Algorithm * Small for gestational age, babies whose birth weight lies below the 10th percentile for that gestational age * Society of Graphic Art, a British arts organization founded in 1919 * Songwriters Guild of America, an organization to help "advance, promote, and benefit" the profession of songwriters * Southern Governors' Association * Standard Galactic Alphabet, the writing system in the ''Commander Keen'' fictional universe * '' Stargate Atlantis'', an American-Canadian science fiction television series and a spin-of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Étale Sheaf , or online commerce
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In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts: * Étale morphism ** Formally étale morphism * Étale cohomology * Étale topology * Étale fundamental group * Étale group scheme * Étale algebra Other * Étale (mountain) in Savoie and Haute-Savoie, France See also * Étalé space * Etail Online shopping is a form of electronic commerce which allows consumers to directly buy goods or services from a seller over the Internet using a web browser or a mobile app. Consumers find a product of interest by visiting the website of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Image With Compact Support
In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations. Definition Let ''f'': ''X'' → ''Y'' be a continuous mapping of locally compact Hausdorff topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on a topological space. The direct image with compact (or proper) support is the functor :''f''!: Sh(''X'') → Sh(''Y'') that sends a sheaf ''F'' on ''X'' to the sheaf ''f''!(''F'') given by the formula :''f''!(''F'')(''U'') := for every open subset ''U'' of ''Y.'' Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff. This defines ''f''!(''F'') as a subsheaf of the direct image sheaf ''f''∗(''F''), and the functoriality of this construction then follows from basic properties of the support and the definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology With Compact Support
In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support. Singular cohomology with compact support Let X be a topological space. Then :\displaystyle H_c^\ast(X;R) := \varinjlim_ H^\ast(X,X\setminus K;R) This is also naturally isomorphic to the cohomology of the sub– chain complex C_c^\ast(X;R) consisting of all singular cochains \phi: C_i(X;R)\to R that have compact support in the sense that there exists some compact K\subseteq X such that \phi vanishes on all chains in X\setminus K. Functorial definition Let X be a topological space and p:X\to \star the map to the point. Using the direct image and direct image with compact support functors p_*,p_!:\text(X)\to \text(\star)=\text, one can define cohomology and cohomology with compact support of a sheaf of abelian groups \mathcal on X as :\displaystyle H^i(X,\mathcal)\ = \ R^ip_*\mathcal, :\displaystyle H^i_c(X,\mathca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brian Conrad
Brian Conrad (born November 20, 1970) is an American mathematician and number theorist, working at Stanford University. Previously, he taught at the University of Michigan and at Columbia University. Conrad and others proved the modularity theorem, also known as the Taniyama-Shimura Conjecture. He proved this in 1999 with Christophe Breuil, Fred Diamond and Richard Taylor, while holding a joint postdoctoral position at Harvard University and the Institute for Advanced Study in Princeton, New Jersey. Conrad received his bachelor's degree from Harvard in 1992, where he won a prize for his undergraduate thesis. He did his doctoral work under Andrew Wiles and went on to receive his Ph.D. from Princeton University in 1996 with a dissertation titled ''Finite Honda Systems And Supersingular Elliptic Curves''. He was also featured as an extra in Nova's ''The Proof''. His identical twin brother Keith Conrad, also a number theorist, is a professor at the University of Connecticut. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
