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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 prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 that has a local presentation, that is, every point in X has an open neighborhood U in which there is an ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Madhav V
Mādhava means Krishna. 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 *Madhava or Madhava-kara, an Indian physician of the 7th or early 8th century *Madhava, titular protagonist of the ancient Indian drama Mālatīmādhava by Bhavabhuti *Madhava, a character in the 11th-century Indian story collection '' Shringara-manjari-katha'' See also * * * Madhavan (other) * Madhavi (other) * Madhab (other) * Madhavaram (other) * ''Madhav'' ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 of 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 sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 one of the most influential mathematicians of the twentieth century. His influence is due both to his original contributions to a remarkably broad spectrum of mathematical theories, and to the mark he left on mathematical practice and style, through some of his own works as well as through the Bourbaki group, of which he was one of the principal founders. 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 Mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Angelo Vistoli
Angelo Vistoli (born 1 June 1958, Massa Lombarda) is an Italian mathematician working on algebraic geometry. Career Vistoli earned a laurea (the Italian equivalent of a master's degree) in 1981 at the University of Bologna. He became a student of Michael Artin at the Massachusetts Institute of Technology, where he completed his Ph.D. in 1987 with the dissertation ''Duality in the Intersection Theory of Moduli Spaces''. After becoming Benjamin Pierce Lecturer and assistant professor in mathematics at Harvard University, he returned to Italy as a professor at the University of Basilicata in 1990. He moved to the University of Bologna in 1993, and to the Scuola Normale Superiore di Pisa in 2006, where he is professor of geometry. In 2024, he succeeded Andrea Ferrara as president of the Classe di Scienze (Faculty of Science) of the Scuola Normale Superiore The Scuola Normale Superiore (commonly known in Italy as "la Normale") is a public university in Pisa and Florence, Tuscany, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. So in the algebraic structure of groups, H is a subquotient of G if there exists a subgroup G' of G and a normal subgroup G'' of G' so that H is isomorphic to G'/G''. 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". As in the context of subgroups, in the context of subquotients the term ''trivial'' may be used for the two subquotients G and \ which are present in every group 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 Example There are subquot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 of Nori semistable vector bundles contains the Tannakian category of essentially finite vector bundles, whose naturally associated group scheme is the fundamental group scheme \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 is ''Nori semistable'' if for any smooth and proper curve C over \bar k and any morphism j:C\to X the pull back j^*(V) is semistable of degree 0. Difference with Nori's original definition Nori semistable vector bundles were called by Nori ''semistable'' causing a lot of confusion with the already ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and 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'' (EGA); 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. Schemes elaborate the fundamental idea that an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. 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 field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''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 element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finite Morphism
In algebraic geometry, a finite morphism between two Affine variety, affine varieties 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 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'': ''X'' → ''Y'' of scheme (mathematics), schemes is a finite morphism if ''Y'' has an open cover by affine schemes :V_i = \mbox \; B_i such that for each ''i'', :f^(V_i) = U_i is an open affine subscheme Spec ''A''''i'', and the restrictio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Group Scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of 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 of group schemes is somewhat better behaved than that of group varieties, since all homomorphisms have 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 initial development of the theory of group schemes was due to Alexander Grothendieck, Michel Rayn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
