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Fine Sheaf
In mathematics, injective sheaves of abelian groups are used to construct the Resolution (algebra), resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext functor, Ext). There is a further group of related concepts applied to sheaf (mathematics), sheaves: flabby (''flasque'' in French), fine, soft (''mou'' in French), acyclic. In the history of the subject they were introduced before the 1957 "Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of ''injective object'' sufficed to found the theory. The other classes of sheaves are historically older notions. The abstract framework for defining cohomology and derived functors does not need them. However, in most concrete situations, resolutions by acyclic sheaves are often easier to construct. Acyclic sheaves therefore serve for computational purposes, for example the Leray spectral sequence. Injective sheaves An injective sheaf \mathcal is a sheaf that is an ... [...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] |
Affine Scheme
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with a sheaf of rings. Zariski topology For any ideal I of R, define V_I to be the set of prime ideals containing I. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\big\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows: For f\in R, define D_f to be the set of prime ideals of R not containing f. Then each D_f is an open subset of \operatorname(R), and \big\ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: In fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning, \operatorname(R) is not, in general, a T1 space. However, \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a function , the codomain is the image of the function's domain . It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms '' injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Subset
In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member 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'' subset can be open except the space itself and the empty set (the indiscrete topology). In pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Lemma
In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed ''p''-form on an open ball in R''n'' is exact for ''p'' with . The lemma was introduced by Henri Poincaré in 1886. Informal Discussion Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in \mathbb^n is exact. In simpler terms, it means that if a differential form is closed in a region that can be shrunk to a point, then it can be written as the derivative of another form; i.e. if dα = 0 on a simplely connected region, we can always find α = dβ; therefore we have d(dβ) = 0, expressed simply as d2 = 0. This concept is used in mathematical physics, particularly in the context of electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is always empty, i.e. if you have a surface (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) \, dx is an example of a 1-form, -form, and can be integral, integrated over an interval [a,b] contained in the domain of f: \int_a^b f(x)\,dx. Similarly, the expression f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz is a -form that can be integrated over a Surface (mathematics), surface S: \int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right). The symbol \wedge denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form f(x,y,z) \, dx \wedge dy \wedge dz represents a volume element that can be integrated over a region of space. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Sheaf
In mathematics, the constant sheaf on a topological space X associated to a set (mathematics), set A is a Sheaf (mathematics), sheaf of sets on X whose stalk (sheaf), stalks are all equal to A. It is denoted by \underline or A_X. The constant presheaf with value A is the presheaf that assigns to each open set, open subset of X the value A, and all of whose restriction maps are the identity map A\to A. The constant sheaf associated to A is the sheafification of the constant presheaf associated to A. This sheaf may be identified with the sheaf of locally constant A-valued functions on X. In certain cases, the set A may be replaced with an object (category theory), object A in some category (mathematics), category \textbf (e.g. when \textbf is the category of abelian groups, or category of commutative rings, commutative rings). Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology. Basics Let X be a topological space, and A a set. The sections o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom. Definitions Points x and y in a topological space X can be '' separated by neighbourhoods'' if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\varnothing). X is a Hausdorff space if any two distinct points in X are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
