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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Closed
In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied: * E is the intersection of an open set and a closed set in X. * For each point x\in E, there is a neighborhood U of x such that E \cap U is closed in U. * E is an open subset of its closure \overline. * The set \overline\setminus E is closed in X. * E is the difference of two closed sets in X. * E is the difference of two open sets in X. The second condition justifies the terminology ''locally closed'' and is Bourbaki's definition of locally closed. To see the second condition implies the third, use the facts that for subsets A \subseteq B, A is closed in B if and only if A = \overline \cap B and that for a subset E and an open subset U, \overline \cap U = \overline \cap U. Examples The interval (0, 1] = (0, 2) \cap , 1/math> is a locally closed subset of \Reals. For another example, consider the relative interior D of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact and without boundary), then the ''k''th cohomology group of ''M'' is isomorphic to the (n-k)th homology group of ''M'', for all integers ''k'' :H^k(M) \cong H_(M). Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation. History A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The ''k''th and (n-k)th Betti numbers of a closed (i.e., compact and without boundary) orientable ''n''-manifold are equal. The ''cohomology'' concept was at that time about 40 y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Image With Compact Support
In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Grothendieck's six operations. Definition Let ''f'': ''X'' → ''Y'' be a continuous mapping of locally compact Hausdorff topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on a topological space. The direct image with compact (or proper) support is the functor :''f''!: Sh(''X'') → Sh(''Y'') that sends a sheaf ''F'' on ''X'' to the sheaf ''f''!(''F'') given by the formula :''f''!(''F'')(''U'') := for every open subset ''U'' of ''Y.'' Here, the notion of a proper map of spaces is unambiguous since the spaces in question are locally compact Hausdorff. This defines ''f''!(''F'') as a subsheaf of the direct image sheaf ''f''∗(''F''), and the functoriality of this construction then follows from basic properties of the support and the definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Twist
In number theory and algebraic geometry, the Tate twist, 'The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist named after John Tate, is an operation on Galois modules. For example, if ''K'' is a field, ''GK'' is its absolute Galois group, and ρ : ''GK'' → AutQ''p''(''V'') is a representation of ''GK'' on a finite-dimensional vector space ''V'' over the field Q''p'' of ''p''-adic numbers, then the Tate twist of ''V'', denoted ''V''(1), is the representation on the tensor product ''V''⊗Q''p''(1), where Q''p''(1) is the ''p''-adic cyclotomic character (i.e. the Tate module of the group of roots of unity in the separable closure ''Ks'' of ''K''). More generally, if ''m'' is a positive integer, the ''m''th Tate twist of ''V'', denoted ''V''(''m''), is the tensor product of ''V'' with the ''m''-fold tensor product of Q''p''(1). Denoting by Q''p''(−1) the dual representation In mathematics, if is a group and is a linear representation of it on the vector space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it is the determinant bundle of holomorphic ''n''-forms on ''V''. This is the dualising object for Serre duality on ''V''. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor ''K'' on ''V'' giving rise to the canonical bundle — it is an equivalence class for linear equivalence on ''V'', and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −''K'' with ''K'' canonical. The anticanonical bundle is the corresponding inverse bundle ω−1. When the anticanonical bundle of V is ample, V is called a Fano variety. The adjunction formula Suppose that ''X'' is a smooth variety and that ''D'' is a smooth divisor on ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation Sheaf
In the mathematical field of algebraic topology, the orientation sheaf on a manifold ''X'' of dimension ''n'' is a locally constant sheaf ''o''''X'' on ''X'' such that the stalk of ''o''''X'' at a point ''x'' is :o_ = \operatorname_n(X, X - \) (in the integer coefficients or some other coefficients). Let \Omega^k_M be the sheaf of differential ''k''-forms on a manifold ''M''. If ''n'' is the dimension of ''M'', then the sheaf :\mathcal_M = \Omega^n_M \otimes \mathcal_M is called the sheaf of (smooth) densities on ''M''. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map: :\textstyle \int_M: \Gamma_c(M, \mathcal_M) \to \mathbb. If ''M'' is oriented; i.e., the orientation sheaf of the tangent bundle of ''M'' is literally trivial, then the above reduces to the usual integration of a differential form. See also * Orientati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Image Functor
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map f : X \to Y, the inverse image functor is a functor from the category of sheaves on ''Y'' to the category of sheaves on ''X''. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features. Definition Suppose we are given a sheaf \mathcal on Y and that we want to transport \mathcal to X using a continuous map f\colon X\to Y. We will call the result the ''inverse image'' or pullback sheaf f^\mathcal. If we try to imitate the direct image by setting :f^\mathcal(U) = \mathcal(f(U)) for each open set U of X, we immediately run into a problem: f(U) is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, we de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Immersion
Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YFriday album), 2001 * ''Open'' (Shaznay Lewis album), 2004 * ''Open'' (Jon Anderson EP), 2011 * ''Open'' (Stick Men album), 2012 * ''Open'' (The Necks album), 2013 * ''Open'', a 1967 album by Julie Driscoll, Brian Auger and the Trinity * ''Open'', a 1979 album by Steve Hillage * "Open" (Queensrÿche song) * "Open" (Mýa song) * "Open", the first song on The Cure album ''Wish'' Literature * ''Open'' (Mexican magazine), a lifestyle Mexican publication * ''Open'' (Indian magazine), an Indian weekly English language magazine featuring current affairs * ''OPEN'' (North Dakota magazine), an out-of-print magazine that was printed in the Fargo, North Dakota area of the U.S. * Open: An Autobiography, Andre Agassi's 2009 memoir Computin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Morphism
In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology. The word ''étale'' is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle. Definition Let \phi : R \to S be a ring homomorphism. This makes S an R-algebra. Choose a monic polynomial f in R /math> and a polynomial g in R /math> such that the derivative f' of f is a unit in (R fR _g. We say that \phi is ''standard étale'' if f and g can be chos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-finite Morphism
In algebraic geometry, a branch of mathematics, a morphism ''f'' : ''X'' → ''Y'' of schemes is quasi-finite if it is of finite type and satisfies any of the following equivalent conditions: * Every point ''x'' of ''X'' is isolated in its fiber ''f''−1(''f''(''x'')). In other words, every fiber is a discrete (hence finite) set. * For every point ''x'' of ''X'', the scheme is a finite κ(''f''(''x'')) scheme. (Here κ(''p'') is the residue field at a point ''p''.) * For every point ''x'' of ''X'', \mathcal_\otimes \kappa(f(x)) is finitely generated over \kappa(f(x)). Quasi-finite morphisms were originally defined by Alexander Grothendieck in SGA 1 and did not include the finite type hypothesis. This hypothesis was added to the definition in EGA II 6.2 because it makes it possible to give an algebraic characterization of quasi-finiteness in terms of stalks. For a general morphism and a point ''x'' in ''X'', ''f'' is said to be quasi-finite at ''x'' if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
