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Essentially Finite Vector Bundle
In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav V. Nori, as the main tool in the construction of the fundamental group scheme. Even if the definition is not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural objects to study in algebraic geometry. The following notion of ''finite vector bundle'' is due to André Weil and will be needed to define essentially finite vector bundles: Finite vector bundles Let X be a scheme and V a vector bundle on X. For f = a_0 + a_1 x + \ldots + a_n x^n \in \mathbb_ /math> an integral polynomial with nonnegative coefficients define :f(V) := \mathcal_X^ \oplus V^ \oplus \left(V^\right)^ \oplus \ldots \oplus \left(V ^\right)^ Then V is called finite if there are two distinct polynomials f,g\in \mathbb_ /math> for which f(V) is isomorphic to g(V). Definition The following two definitions coincide whenever X is a reduced, connected and prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle (algebraic Geometry)
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] |
Madhav V
Mādhava means Lord Krishna an incarnation of Vishnu. It may also refer to: *a Sanskrit patronymic, "descendant of Madhu (a man of the Yadu tribe)". ** especially of Krishna, see Madhava (Vishnu) *** an icon of Krishna ** Madhava of Sangamagrama, fourteenth-century Indian mathematician ** Madhvacharya, philosopher in the Vaishnavism tradition ** Madhava Vidyaranya, Advaita saint and brother of Sayana ** Venkata Madhava, 10th to 12th century commentator of the Rigveda ** Madhavdeva, 16th-century proponent of Ekasarana dharma, neo-Vaishnavism of Assam *relating to springtime; the first month of spring, see Chaitra *a name of Krishna *Madhava or Madhava-kara, an Indian physician of the 7th or early 8th century See also *Madhavan (other) *Madhavi (other) *Magha (month) Maagha (Hindi: माघ ''maagh'') is a month of the Hindu calendar. In India's national civil calendar, it's the eleventh month of the year, corresponding to January/February in the Gregorian c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group Scheme
In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental group. Although its existence was conjectured by Alexander Grothendieck, the first proof if its existence is due, for schemes defined over fields, to Madhav Nori. A proof of its existence for schemes defined over Dedekind schemes is due to Marco Antei, Michel Emsalem and Carlo Gasbarri. History The (topological) fundamental group associated with a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. Although it is still being studied for the classification of algebraic varieties even in algebraic geometry, for many applications the fundamental group has been found to be inadequate for the classification of objects, such as schemes, that are more than just topological spaces. The same ... [...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] |
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] |
Angelo Vistoli
Angelo Vistoli (born June 1, 1958, Massa Lombarda) is an Italian mathematician famous for his work in Algebraic Geometry. Career Angelo Vistoli is currently professor of geometry at the Scuola Normale Superiore di Pisa, an honor reserved for few mathematicians in Italy and the whole world. Indeed, Vistoli is among the most influential algebraic geometers in Italy and the world, as evidenced by his long list of publications among the world's most prestigious journals, including Annals of Mathematics and Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors .... One of his most influential paper is indeed ''Intersection theory on algebraic stacks and on their moduli spaces''. References {{DEFAULTSORT:Vistoli, Angelo 20th-century Italian mathematicians 21st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subquotient
In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory. In the literature about sporadic groups wordings like «H is involved in G» can be found with the apparent meaning of «H is a subquotient of G». A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e.g., Harish-Chandra's subquotient theorem. p. 310 Examples Of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 as "pariah groups". Order relation The relation ''subquotient of'' is an order relation. Proof of transitivity for groups Let H'/H'' be subquotient of H, furthermore H := G'/G'' be subquotient of G and \varphi \colon G' \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nori-semistable
In mathematics, a Nori semistable vector bundle is a particular type of vector bundle whose first definition has been first implicitly suggested by Madhav V. Nori, as one of the main ingredients for the construction of the fundamental group scheme. The original definition given by Nori was obviously not called ''Nori semistable''. Also, Nori's definition was different from the one suggested nowadays. The Category (mathematics), category of Nori semistable vector bundles contains the Tannakian formalism, Tannakian category of essentially finite vector bundles, whose naturally associated group scheme is the fundamental group scheme In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental grou ... \pi_1(X,x). Definition Let X be a scheme over a field k and V a vector bundle on X. It is said that V i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Morphism
In algebraic geometry, a finite morphism between two affine varieties In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime idea ... X, Y is a dense Regular map (algebraic geometry), regular map which induces isomorphic inclusion k\left[Y\right]\hookrightarrow k\left[X\right] between their Coordinate ring, coordinate rings, such that k\left[X\right] is integral over k\left[Y\right]. This definition can be extended to the quasi-projective varieties, such that a Regular map (algebraic geometry), regular map f\colon X\to Y between quasiprojective varieties is finite if any point like y\in Y has an affine neighbourhood V such that U=f^(V) is affine and f\colon U\to V is a finite map (in view of the previous definition, because it is between affine varieties). Definition by Schemes A morphism ''f'': ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The Category (mathematics), category of group schemes is somewhat better behaved than that of Group variety, group varieties, since all homomorphisms have Kernel (category theory), kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The ini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
