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Flag Bundle
In algebraic geometry, the flag bundle of a flagHere, E_i is a subbundle not subsheaf of E_. :E_: E = E_l \supsetneq \cdots \supsetneq E_1 \supsetneq 0 of vector bundles on an algebraic scheme ''X'' is the algebraic scheme over ''X'': :p: \operatorname(E_) \to X such that p^(x) is a flag V_ of vector spaces such that V_i is a vector subspace of (E_i)_x of dimension ''i''. If ''X'' is a point, then a flag bundle is a flag variety and if the length of the flag is one, then it is the Grassmann bundle In algebraic geometry, the Grassmann ''d''-plane bundle of a vector bundle ''E'' on an algebraic scheme ''X'' is a scheme over ''X'': :p: G_d(E) \to X such that the fiber p^(x) = G_d(E_x) is the Grassmannian of the ''d''-dimensional vector subspace ...; hence, a flag bundle is a common generalization of these two notions. Construction A flag bundle can be constructed inductively. References * *Expo. VI, § 4. of Algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Scheme
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] |
Flag Variety
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space ''V'' over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmann Bundle
In algebraic geometry, the Grassmann ''d''-plane bundle of a vector bundle ''E'' on an algebraic scheme ''X'' is a scheme over ''X'': :p: G_d(E) \to X such that the fiber p^(x) = G_d(E_x) is the Grassmannian of the ''d''-dimensional vector subspaces of E_x. For example, G_1(E) = \mathbb(E) is the projective bundle of ''E''. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme. Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle ''S'' and universal quotient bundle ''Q'' that fit into :0 \to S \to p^*E \to Q \to 0. Specifically, if ''V'' is in the fiber ''p''−1(''x''), then the fiber of ''S'' over ''V'' is ''V'' itself; thus, ''S'' has rank ''r'' = ''d'' = dim(''V'') and \wedge^d S is the determinant line bundle. Now, by the universal property of a projective bundle, the injection \wedg ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandre Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He received the Fields Medal in 1966 for advances in algebraic geometry, homological algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luc Illusie
Luc Illusie (; born 1940) is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences. Biography Luc Illusie entered the École Normale Supérieure in 1959. At first a student of the mathematician Henri Cartan, he participated in the Cartan–Schwartz seminar of 1963–1964. In 1964, following Cartan's advice, he began to work with Alexandre Grothendieck, collaborating with him on two volumes of the latter's Séminaire de Géométrie Algébrique du Bois Marie. In 1970, Illusie introduced the concept of the cotangent complex. A researcher in the Centre national de la recherche scientifique from 1964 to 1976, Illusie then became a professor at the University of Paris-Sud, retiring as emeritus professor in 2005. Between 1984 and 1995, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
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, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |