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Reflexive Sheaf
In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a reflexive sheaf is a locally free sheaf of finite rank. The notion is important both in scheme theory and complex algebraic geometry. For the theory of reflexive sheaves, one works over an integral noetherian scheme. A reflexive sheaf is torsion-free. The dual of a coherent sheaf is reflexive. Usually, the product of reflexive sheaves is defined as the reflexive hull of their tensor products (so the result is reflexive). A coherent sheaf ''F'' is said to be "normal" in the sense of Barth if the restriction F(U) \to F(U - Y) is bijective for every open subset ''U'' and a closed subset ''Y'' of ''U'' of codimension at least 2. With this terminology, a coherent sheaf on an integral normal scheme is reflexive if and only if it is torsion-fre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Torsion-free Sheaf
In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule contains only the zero element. In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module. Examples of torsion-free modules Over a commutative ring ''R'' with total quotient ring ''K'', a module ''M'' is torsion-free if and only if Tor1(''K''/''R'',''M'') vanishes. Therefore flat modules, and in particular free and projective modules, ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion Sheaf
In mathematics, a torsion sheaf is a sheaf of abelian groups \mathcal on a site for which, for every object ''U'', the space of sections \Gamma(U, \mathcal) is a torsion abelian group. Similarly, for a prime number ''p'', we say a sheaf \mathcal is ''p''-torsion if every section over any object is killed by a power of ''p''. A torsion sheaf on an étale site is the union of its constructible subsheaves. See also * Twisted sheaf In mathematics, a twisted sheaf is a variant of a coherent sheaf. Precisely, it is specified by: an open covering in the étale topology ''U'i'', coherent sheaves ''F'i'' over ''U'i'', a Čech 2-cocycle ''θ'' for \mathbb_m on the coverin ... Notes References * * J. S. Milne, ''Étale Cohomology'' * Sheaf theory {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsionless Module
In abstract algebra, a module (mathematics), module ''M'' over a ring (mathematics), ring ''R'' is called torsionless if it can be embedded into some direct product ''R''''I''. Equivalently, ''M'' is torsionless if each non-zero element of ''M'' has non-zero image under some ''R''-linear functional ''f'': : f\in M^=\operatorname_R(M,R),\quad f(m)\ne 0. This notion was introduced by Hyman Bass. Properties and examples A module is torsionless if and only if the canonical map into its double dual module, dual, : M\to M^=\operatorname_R(M^,R), \quad m\mapsto (f\mapsto f(m)), m\in M, f\in M^, is injective. If this map is bijective then the module is called reflexive. For this reason, torsionless modules are also known as semi-reflexive. * A unital free module is torsionless. More generally, a direct sum of torsionless modules is torsionless. * A free module is reflexive if it is finitely generated module, finitely generated, and for some rings there are also infinitely genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dualizing Sheaf
In algebraic geometry, the dualizing sheaf on a proper scheme ''X'' of dimension ''n'' over a field ''k'' is a coherent sheaf \omega_X together with a linear functional :t_X: \operatorname^n(X, \omega_X) \to k that induces a natural isomorphism of vector spaces :\operatorname_X(F, \omega_X) \simeq \operatorname^n(X, F)^*, \, \varphi \mapsto t_X \circ \varphi for each coherent sheaf ''F'' on ''X'' (the superscript * refers to a dual vector space). The linear functional t_X is called a trace morphism. A pair (\omega_X, t_X), if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, \omega_X is an object representing the contravariant functor F \mapsto \operatorname^n(X, F)^* from the category of coherent sheaves on ''X'' to the category of ''k''-vector spaces. For a normal projective variety ''X'', the dualizing sheaf exists and it is in fact the canonical sheaf: \omega_X = \mathcal_X(K_X) where K_X is a canonical divisor. More generally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Sheaf
The adjective canonical is applied in many contexts to mean 'according to the canon' the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, ''canonical example'' is often used to mean 'archetype'. Science and technology * Canonical form, a natural unique representation of an object, or a preferred notation for some object Mathematics * * Canonical coordinates, sets of coordinates that can be used to describe a physical system at any given point in time * Canonical map, a morphism that is uniquely defined by its main property * Canonical polyhedron, a polyhedron whose edges are all tangent to a common sphere, whose center is the average of its vertices * Canonical ring, a graded ring associated to an algebraic variety * Canonical injection, in set theory * Canonical representative, in set theory a standard member of each element of a set partition Differential geometry * Canonical one-fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conductor (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] |
Normal Scheme
In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and only if the ring ''O''(''X'') of regular functions on ''X'' is an integrally closed domain. A variety ''X'' over a field is normal if and only if every finite birational morphism from any variety ''Y'' to ''X'' is an isomorphism. Normal varieties were introduced by . Geometric and algebraic interpretations of normality A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve ''X'' in the affine plane ''A''2 defined by ''x''2 = ''y''3 is not normal, because there is a finite birational morphism ''A''1 → ''X'' (namely, ''t'' maps to (''t''3, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent 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 that has a local presentation, that is, every point in X has an open neighborhood U in which there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Noetherian Scheme
In algebraic geometry, a Noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, where each A_i is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and only if it is locally Noetherian and compact. As with Noetherian rings, the concept is named after Emmy Noether. It can be shown that, in a locally Noetherian scheme, if \operatorname A is an open affine subset, then ''A'' is a Noetherian ring; in particular, \operatorname A is a Noetherian scheme if and only if ''A'' is a Noetherian ring. For a locally Noetherian scheme ''X,'' the local rings \mathcal_ are also Noetherian rings. A Noetherian scheme is a Noetherian topological space. But the converse is false in general; consider, for example, the spectrum of a non-Noetherian valuation ring. The definitions extend to formal schemes. Properties and Noetherian hypotheses Having ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
