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Image Functors For Sheaves
In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses. Given a continuous mapping ''f'': ''X'' → ''Y'' of topological spaces, and the category Sh(–) of sheaves of abelian groups on a topological space. The functors in question are * direct image ''f''∗ : Sh(''X'') → Sh(''Y'') * inverse image ''f''∗ : Sh(''Y'') → Sh(''X'') * direct image with compact support ''f''! : Sh(''X'') → Sh(''Y'') * exceptional inverse image ''Rf''! : ''D''(Sh(''Y'')) → ''D''(Sh(''X'')). The exclamation mark is often pronounced " shriek" (slang for exclamation mark), and the maps called "''f'' shriek" or "''f'' lower shriek" and "''f'' upper shriek"—see also shriek map. The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to étale sheaves on schemes. Adjointness The fu ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
étale Topology
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale topology was originally introduced by Grothendieck to define étale cohomology, and this is still the étale topology's most well-known use. Definitions For any scheme ''X'', let Ét(''X'') be the category of all étale morphisms from a scheme to ''X''. This is the analog of the category of open subsets of ''X'' (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of ''X''. The intersection of two objects corresponds to their fiber product over ''X''. Ét(''X'') is a large category, meaning that its objects do not form a set. An étale presheaf on ''X'' is a contravariant functor from Ét(''X'') to the category of sets. A presheaf ... [...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 international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Six Operations
In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes . The basic insight was that many of the elementary facts relating cohomology on ''X'' and ''Y'' were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as ''D''-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives. The operations The operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors. * the direct image f_* * the inverse image f^* * the proper (or extraordinary) direct image f_! * the proper (or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distinguished Triangle
In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space ''X'' to complexes of sheaves, viewed as objects of the derived category of sheaves on ''X''. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
Open Subset
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base Change Map
In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: :g^*(R^r f_* \mathcal) \to R^r f'_*(g'^*\mathcal) where :\begin X' & \stackrel\to & X \\ f' \downarrow & & \downarrow f \\ S' & \stackrel g \to & S \end is a Cartesian square of topological spaces and \mathcal is a sheaf on ''X''. Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps ''f'', in algebraic geometry for (quasi-)coherent sheaves and ''f'' proper or ''g'' flat, similarly in analytic geometry, but also for étale sheaves for ''f'' proper or ''g'' smooth. Introduction A simple base change phenomenon arises in commutative algebra when ''A'' is a commutative ring and ''B'' and ''A' ''are two ''A''-algebras. Let B' = B \otimes_A A'. In this situation, given a ''B''-module ''M'', there is an isomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perverse Sheaf
The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces (intersection homology of Mark Goresky and Robert MacPherson) and the algebraic theory of differential equations ( microlocal calculus and holonomic D-modules of Joseph Bernstein, Masaki Kashiwara and Takahiro Kawai). It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Verdier Duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism. Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves. Verdier duality Verdier duality states that (subject to suitable finiteness conditions disc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
