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Picard Group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral Scheme (mathematics), schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the Spectrum of a ring, spectrum of a Dedekind domain is its ''ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Domain
In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism :X \times Y \to Y is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic geometry: a topological space is compact if and only if the above projection map is closed with respect to topological products. The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete. A complex variety is complete if and only if it is compact as a complex-analytic variety. The most common example of a complete variety is a projective variety, but there do exist complete non-projective varieties in dimensions 2 and higher. While any complete nonsingular surface is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective. The first examples of non-projective complete va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Theory Of Abelian Varieties
In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''k''. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions. Definition Let ''A'' be an abelian variety over a field ''k''. We define \operatorname^0 (A) \subset \operatorname (A) to be the subgroup of the Picard group consisting of line bundles ''L'' such that m^*L \cong p^*L \otimes q^*L, where m, p, q are the multiplication and projection maps A \times_k A \to A respectively. An element of \operatorname^0(A) is called a degree 0 line bundle on ''A''. To ''A'' one then associates a dual abelian variety ''A''v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a ''k''-variety ''T'' is defined to be a line bundle ''L'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representable Functor
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings. From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of upper sets in posets, and Yoneda's representability theorem generalizes Cayley's theorem in group theory. Definition Let C be a locally small category and let Set be the category of sets. For each object ''A'' of C let Hom(''A'',–) be the hom functor that maps object ''X'' to the set Hom(''A'',''X''). A functor ''F'' : C → Set is said to be representable if it is naturally isomorphic to Hom(''A'',–) for some object ''A'' of C. A representation of ''F'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dolbeault Cohomology
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohomology groups H^(M, \Complex) depend on a pair of integers ''p'' and ''q'' and are realized as a subquotient of the space of complex differential forms of degree (''p'',''q''). Construction of the cohomology groups Let Ω''p'',''q'' be the vector bundle of complex differential forms of degree (''p'',''q''). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections :\bar:\Omega^\to\Omega^ Since :\bar^2=0 this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space :H^(M,\Complex)=\frac . Dolbeault cohomology of vector bundles If ''E'' is a holomorphic vector bundle on a complex manifold ''X'', then one can define likewise a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Sequence
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let ''M'' be a complex manifold, and write ''O''''M'' for the sheaf of holomorphic functions on ''M''. Let ''O''''M''* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism :\exp : \mathcal O_M \to \mathcal O_M^*, because for a holomorphic function ''f'', exp(''f'') is a non-vanishing holomorphic function, and exp(''f'' + ''g'') = exp(''f'')exp(''g''). Its kernel is the sheaf 2π''i''Z of locally constant functions on ''M'' taking the values 2π''in'', with ''n'' an integer. The exponential sheaf sequence is therefore :0\to 2\pi i\,\mathbb Z \to \mathcal O_M\to\mathcal O_M^*\to 0. The exponential mapping here is not always a surjective map on sections; this can be seen for example when ''M'' is a punctured disk in the complex plane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Affine Space
Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces (that is, vector spaces) in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps."* Accordingly, a complex affine space, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin. Affine geometry is one of the two main branches of classical algebraic geometry, the other being projective geometry. A complex affine space can be obtained from a complex projective space by fixing a hyperplane, which can be thought of as a hyperplane of ideal points "at infinity" of the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
