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Constructible Set (topology)
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] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Noetherian Topological Space
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that ''every'' subset is compact. Definition A topological space X is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence : Y_1 \supseteq Y_2 \supseteq \cdots of closed subsets Y_i of X, there is an integer m such that Y_m=Y_=\cdots. Properties * A topological space X is Noetherian if and only if every subspace of X is compact (i.e., X is hereditarily compact), and if and only if every open subset of X is c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups. Biography He studied at the ETH Zürich, where he came under the influence of the topologist Heinz Hopf and Lie-group theorist Eduard Stiefel. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of Jean Leray and Henri Cartan. With Hirzebruch, he significantly developed the theory of characteristic classes in the early 1950s. He collaborated with Jacques Tits in fundamental work on algebraic groups, and with Harish-Chandra on their arithmetic subgroups. In an algebraic group ''G'' a ''Borel subgroup'' ''H'' is one mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Sheaf
In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in étale cohomology constructible sheaves are defined in a similar way . For the derived category of constructible sheaves, see a section in ℓ-adic sheaf. The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible. Definition of étale constructible sheaves on a scheme ''X'' Here we use the definition of constructible étale sheaves from the book by Freitag and Kiehl referenced below. In what follows in this subsection, all sheaves \mathcal on schemes X are étale sheaves unless otherwise noted. A sheaf \mathcal is called constructible if X can be written as a finite union of locally closed subschemes i_Y:Y \to X such that for each subsche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Topology
In commutative algebra, the constructible topology on the spectrum \operatorname(A) of a commutative ring A is a topology where each closed set is the image of \operatorname (B) in \operatorname(A) for some algebra ''B'' over ''A''. An important feature of this construction is that the map \operatorname(B) \to \operatorname(A) is a closed map with respect to the constructible topology. With respect to this topology, \operatorname(A) is a compact, Hausdorff, and totally disconnected topological space (i.e., a Stone space). In general, the constructible topology is a finer topology than the Zariski topology, and the two topologies coincide if and only if A / \operatorname(A) is a von Neumann regular ring, where \operatorname(A) is the nilradical of ''A''. Despite the terminology being similar, the constructible topology is not the same as the set of all constructible sets. See also *Constructible set (topology) In topology, constructible sets are a class of subsets of a topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e., :f_P\colon \mathcal_ \to \mathcal_ is a flat map for all ''P'' in ''X''. A map of rings A\to B is called flat if it is a homomorphism that makes ''B'' a flat ''A''-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. Two basic intuitions regarding flat morphisms are: *flatness is a generic property; and *the failure of flatness occurs on the jumping set of the morphism. The first of these comes from commutative algebra: subject to some finiteness conditions on ''f'', it can be shown that there is a non-empty open subscheme Y' of ''Y'', such that ''f'' restricted to ''Y''′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to ''f'' and the inclusio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasicoherent Sheaf
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] |
Glossary Of Algebraic Geometry
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B. Image and inverse image may also be defined for general binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a function from the set X to the set Y. Image of an element If x is a member of X, then the image of x under f, denoted f(x), is the value of f when applied to x. f(x) is alternatively known as the output of f for argument x. Given y, the function f is said to "" or "" if there exists some x in the function's domain such that f(x) = y. Similarly, given a set S, f is said to "" if there exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stacks Project
The Stacks Project is an open source collaborative mathematics textbook writing project with the aim to cover "algebraic stacks and the algebraic geometry needed to define them". , the book consists of 115 chapters (excluding the license and index chapters) spreading over 7500 pages. The maintainer of the project, who reviews and accepts the changes, is Aise Johan de Jong. See alsoKerodona Stacks project inspired online textbook on categorical homotopy theory maintained by Jacob Lurie Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow. Life When he was a student in the Science, Mathematics, and Computer Science ... References External linksProject website*Latest from the Stacks Project(as of 2013) (Accessed 2020-04-01) Mathematics textbooks {{mathematics-lit-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Éléments De Géométrie Algébrique
The ''Éléments de géométrie algébrique'' ("Elements of algebraic geometry, Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French language, French, on algebraic geometry that was published (in eight parts or fascicle (book), fascicles) from 1960 through 1967 by the ''Institut des Hautes Études Scientifiques''. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of Scheme (mathematics), schemes, which he defined. The work is now considered the foundation stone and basic reference of modern algebraic geometry. Editions Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the ''Séminaire de géométrie algébrique'' (known as ''SGA''). Indeed, as explained by G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
