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Koszul Duality
In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohomology). The prototypical example of Koszul duality was introduced by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand,. It establishes a duality between the derived category of a symmetric algebra and that of an exterior algebra, as well as the BGG correspondence, which links the stable category of finite-dimensional graded modules over an exterior algebra to the bounded derived category of coherent sheaves on projective space. The importance of the notion rests on the suspicion that Koszul duality seems quite ubiquitous in nature. Koszul duality for graded modules over Koszul algebras The simplest, and in a sense prototypical case of Koszul duality arises as follows: for a 1-dimensional vector space ''V'' over a field ''k'', with dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BGG Correspondence
In mathematics, the Bernstein-Gelfand-Gelfand correspondence or BGG correspondence for short is the first example of the Koszul duality. Established by Joseph Bernstein, Israel Gelfand, and Sergei Gelfand, the correspondence is an explicit triangulated equivalence that relates the bounded derived category of coherent sheaves on the projective space \mathbb^ and the stable category of graded modules \operatorname \wedge V over the exterior algebra \wedge V; i.e., :D^b(\mathbb) \simeq \overline. In the noncommutative setting, Martínez Villa and Saorín generalized the BGG correspondence to finite-dimensional self-injective Koszul algebras A with coherent sheaf, coherent Koszul duals A^. Roughly speaking, they proved that the stable category of finite-dimensional graded modules over a finite-dimensional self-injective Koszul algebra A is triangulated equivalent to the bounded derived category of the tails category of the Koszul dual A^ (when A^ is coherent). References Furthe ... [...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] |
Quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms :H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bullet) \to H^n(B^\bullet)) of homology groups (respectively, of cohomology groups) are isomorphisms for all ''n''. In the theory of model categories, quasi-isomorphisms are sometimes used as the class of weak equivalences when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of Bousfield localization in homotopy theory. See also * Derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ... References *Gelfand, Sergei I., Manin, Yuri I. ''Methods of Homological Alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dg-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_\bullet of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D-modules
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of ''D''-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. This approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems ( holonomic systems), and on the characteristic variety cut out by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Module
Grade most commonly refers to: * Grading in education, a measurement of a student's performance by educational assessment (e.g. A, pass, etc.) * A designation for students, classes and curricula indicating the number of the year a student has reached in a given educational stage (e.g. first grade, second grade, K–12, etc.) * Grade (slope), the steepness of a slope * Graded voting Grade or grading may also refer to: Music * Grade (music), a formally assessed level of profiency in a musical instrument * Grade (band), punk rock band * Grades (producer), British electronic dance music producer and DJ Science and technology Biology and medicine * Grading (tumors), a measure of the aggressiveness of a tumor in medicine * The Grading of Recommendations Assessment, Development and Evaluation (GRADE) approach * Evolutionary grade, a paraphyletic group of organisms Geology * Graded bedding, a description of the variation in grain size through a bed in a sedimentary rock * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ext Functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The group cohomology, cohomology of groups, Lie algebra cohomology, Lie algebras, and Hochschild cohomology, associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies group extension, extensions of one module (mathematics), module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring (mathematics), ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let R be a ring and let R\text be the category (mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Ring
In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring whose multiplication ∗ is defined by for all in ''R''. The opposite ring can be used to define multimodules, a generalization of bimodules. They also help clarify the relationship between left and right modules (see '). Monoids, groups, rings, and algebras can all be viewed as categories with a single object. The construction of the opposite category generalizes the opposite group, opposite ring, etc. Relation to automorphisms and antiautomorphisms In this section the symbol for multiplication in the opposite ring is changed from asterisk to diamond, to avoid confusing it with some unary operations. A ring is called a ''self-opposite'' ring if it is isomorphic to its opposite ring, which name indicates that R^\text is essentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding universal property (see #Adjunction and universal property, below). The tensor algebra is important because many other algebras arise as quotient associative algebra, quotient algebras of ''T''(''V''). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bi-algebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Algebra
In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras. Definition A graded quadratic algebra ''A'' is determined by a vector space of generators ''V'' = ''A''1 and a subspace of homogeneous quadratic relations ''S'' ⊂ ''V'' ⊗ ''V''. Thus : A=T(V)/\langle S\rangle and inherits its grading from the tensor algebra ''T''(''V''). If the subspace of relations is instead allowed to also contain inhomogeneous degree 2 elements, i.e. ''S'' ⊂ ''k'' ⊕ ''V'' ⊕ (''V'' ⊗ ''V''), this construction results in a filtered quadratic algebra. A graded quadratic algebra ''A'' as above admits a quadratic dual: the quadratic algebra generated by ''V''* and with quadratic relations forming the orthogonal complement of ''S'' in ''V' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koszul Algebra
In abstract algebra, a Koszul algebra R is a graded k-algebra over which the ground field k has a linear minimal graded free resolution, ''i.e.'', there exists an exact sequence: :\cdots \rightarrow (R(-i))^ \rightarrow \cdots \rightarrow (R(-2))^ \rightarrow (R(-1))^ \rightarrow R \rightarrow k \rightarrow 0. for some nonnegative integers b_i. Here R(-j) is the graded algebra R with grading shifted up by j, ''i.e.'' R(-j)_i = R_, and the exponent b_i refers to the b_i-fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, ''e.g'', R = k ,y(xy) . The concept is named after the French mathematician Jean-Lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
