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Coherent Duality
In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point. The resulting theory is now sometimes called Serre–Grothendieck–Verdier duality, and is a basic tool in algebraic geometry. A treatment of this theory, ''Residues and Duality'' (1966) by Robin Hart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre Duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an ''n''-dimensional variety, the theorem says that a cohomology group H^i is the dual space of another one, H^. Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf. The Serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties. In this setting, the Serre duality theorem is an application of Hodge theory for Dolbeault cohomology, and may be seen as a result in the theory of elliptic operators. These two different interpretations of Serre du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only Connected space, connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP''n'' is a closed ''n''-dimensional manifold. The complex projective space CP''n'' is a closed 2''n''-dimensional manifold. A Real line, line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If M is a closed connected n-manifold, the n-th homology group H_(M;\mathbb) is \mathbb or 0 depending on whether M is Orientability, orientable or not. Moreover, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exceptional Inverse Image Functor
In mathematics, more specifically sheaf theory, a branch of topology and algebraic geometry, the exceptional inverse image functor is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form. Definition Let ''f'': ''X'' → ''Y'' be a continuous map of topological spaces or a morphism of schemes. Then the exceptional inverse image is a functor :R''f''!: D(''Y'') → D(''X'') where D(–) denotes the derived category of sheaves of abelian groups or modules over a fixed ring. It is defined to be the right adjoint of the total derived functor R''f''! of the direct image with compact support. Its existence follows from certain properties of R''f''! and general theorems about existence of adjoint functors, as does the unicity. The notation R''f''! is an abuse of notation insofar as there is in general no functor ''f''! whose derived functor would be R''f''!. Examples and properties *If ''f'': ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Adjoint
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects c in \mathcal and d in \mathcal, a bijection between the respective morphism sets :\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gorenstein Ring
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring ). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings. Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings. For Noetherian local rings, there is the following chain of inclusions. Definitions A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring, as de ... [...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] |
Invertible Sheaf
In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Definition Let (''X'', ''O''''X'') be a ringed space. Isomorphism classes of sheaves of ''O''''X''-modules form a monoid under the operation of tensor product of ''O''''X''-modules. The identity element for this operation is ''O''''X'' itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if ''L'' is a sheaf of ''O''''X''-modules, then ''L'' is called invertible if it satisfies any of the following equivalent conditions: Stacks Project, tag 01CR * There exists a sheaf ''M'' such that L \otimes_ M \cong \mathcal_X. * The natural homomorphism L \otimes_ L^\vee \to \mathcal_X is an isomorphism, where L^\vee denote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eben Matlis
Eben Matlis (August 28, 1923 - March 27, 2015) was a mathematician known for his contributions to the theory of rings and modules, especially for his work with injective modules over commutative Noetherian rings, and his introduction of Matlis duality. Matlis earned his Ph.D. at the University of Chicago in 1958, with Irving Kaplansky as advisor. He is an emeritus professor at Northwestern University and was a member of the Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ... from August 1962 to June 1963. Selected works * * * References External links * 1923 births 2015 deaths 20th-century American mathematicians Algebraists University of Chicago alumni Northwestern University faculty Institute for Advanced Study people {{US-mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralf Strebel
Ralph (pronounced or ) is a male name of English origin, derived from the Old English ''Rædwulf'' and Old High German ''Radulf'', cognate with the Old Norse ''Raðulfr'' (''rað'' "counsel" and ''ulfr'' "wolf"). The most common forms are: * Ralph, the common variant form in English, which takes either of the given pronunciations. * Rafe, variant form which is less common; this spelling is always pronounced . * Raif, a very rare variant. Raif Rackstraw from H.M.S. Pinafore * Ralf, the traditional variant form in Dutch, German, Swedish, and Polish. * Ralfs, the traditional variant form in Latvian. * Raoul, the traditional variant form in French. * Raúl, the traditional variant form in Spanish. * Raul, the traditional variant form in Portuguese and Italian. * Raül, the traditional variant form in Catalan. * Rádhulbh, the traditional variant form in Irish. First name Middle Ages * Ralph the Timid (died 1057), pre-Conquest Norman earl of Hereford, England * Ralph de Gael ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Local Duality
In commutative algebra, Grothendieck local duality is a duality theorem for cohomology of modules over local rings, analogous to Serre duality of coherent sheaves. Statement Suppose that ''R'' is a Cohen–Macaulay local ring of dimension ''d'' with maximal ideal ''m'' and residue field ''k'' = ''R''/''m''. Let ''E''(''k'') be a Matlis module, an injective hull of ''k'', and let be the completion of its dualizing module. Then for any ''R''-module ''M'' there is an isomorphism of modules over the completion of ''R'': : \operatorname_R^i(M,\overline\Omega) \cong \operatorname_R(H_m^(M),E(k)) where ''H''''m'' is a local cohomology group. There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a dualizing complex. See also *Matlis duality In algebra, Matlis duality is a duality (mathematics), duality between Artinian module, Artinian and Noetherian module, Noetherian module (mathematics), modules ov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
