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Koszul Complex
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an Regular sequence#Definitions, M-regular sequence, and hence it can be used to prove basic facts about the Depth (ring theory), depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension. Moreover, in certain circumstances, the complex is the complex of Hilbert's syzygy theorem#Syzygies (relations), syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth. Definition Let ''A'' be a commutative ring and ''s: Ar → A'' an ''A''-linear map. Its Koszul complex ''Ks'' is : \bigwedge^r A^r\ \to\ \bigwedge^A^r\ \ ... [...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] |
Differential Graded Algebra
In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geometric space. Explicitly, a differential graded algebra is a graded associative algebra with a chain complex structure that is compatible with the algebra structure. In geometry, the de Rham algebra of differential forms on a manifold has the structure of a differential graded algebra, and it encodes the de Rham cohomology of the manifold. In algebraic topology, the singular cochains of a topological space form a DGA encoding the singular cohomology. Moreover, American mathematician Dennis Sullivan developed a DGA to encode the rational homotopy type of topological spaces. __TOC__ Definitions Let A_\bullet = \bigoplus\nolimits_ A_i be a \mathbb-graded algebra, with product \cdot, equipped with a map d\colon A_\bullet \to A_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (homology Theory)
Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social sciences ** Business cycle, the downward and upward movement of gross domestic product (GDP) around its ostensible, long-term growth trend Arts, entertainment, and media Films * ''Cycle'' (2008 film), a Malayalam film * ''Cycle'' (2017 film), a Marathi film Literature * ''Cycle'' (magazine), an American motorcycling enthusiast magazine * Literary cycle, a group of stories focused on common figures Music Musical terminology * Cycle (music), a set of musical pieces that belong together ** Cyclic form, a technique of construction involving multiple sections or movements ** Interval cycle, a collection of pitch classes generated from a sequence of the same interval class ** Song cycle, individually complete songs des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set and ring , there is a free -module with basis , which is called the ''free module on'' or ''module of formal'' -''linear combinations'' of the elements of . A free abelian group is precisely a free module over the ring \Z of integers. Definition For a ring R and an R- module M, the set E\subseteq M is a basis for M if: * E is a generating set for M; that is to say, every element of M is a finite sum of elements of E multiplied by coefficients in R; and * E is linearly independent: for every set \\subset E of distinct elements, r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M implies that r_1 = r_2 = \cdots = r_n = 0_R (where 0_M is the zero element of M and 0_R is the zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted . An element of the form v \otimes w is called the tensor product of v and w. An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for V and W, a basis of V \otimes W is formed by all tensor products of a basis element of V and a basis element of W. The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space Z factors uniquely through a linear map V\otimes W\to Z (see the section below ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitions (which try to list the objects that a term describes).Lyons, John. "Semantics, vol. I." Cambridge: Cambridge (1977). p.158 and on. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what the mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed. Basic terminology In modern usage, a definition is something, typically expressed in words, that at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilator (ring Theory)
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of . Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case of noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal. Definitions Let ''R'' be a ring, and let ''M'' be a left ''R''- module. Choose a non-empty subset ''S'' of ''M''. The ''annihilator'' of ''S'', denoted Ann''R''(''S''), is the set of all elements ''r'' in ''R'' such that, for all ''s'' in ''S'', . In set notation, :\mathrm_R(S)=\ It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is a torsion set). Subsets of right modules may be used as well, after the modifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume Form
In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the line bundle \textstyle^n(T^*M), denoted as \Omega^n(M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a ''twisted volume form' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-duality
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called ''primal''). Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
