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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] |
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] |
Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; ; ; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory. Biography Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University. In 1981, he emigrated to the United States due to growing antisemitism in the Soviet Union. Bernstein was a professor at Harvard during 1983-1993. He was a visiting scholar at the Institute for Advanced Study in 1985-86 and again in 1997-98. In 1993, he moved to Israel to take a professorship at Tel Aviv University (emeritus since 2014). Awards and honors Bernstein received a gold medal at the 1962 International Mathematical Olympiad. He was elected to the Israel Academy of Sciences and Humanities in 2002 and was elected to the United States National Academy of Sciences in 2004. In 2004, Bernstein was awarded the Israel Prize for mathematics. In 1998, he was an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (, , ; – 5 October 2009) was a prominent Soviet and American mathematician, one of the greatest mathematicians of the 20th century, biologist, teacher and organizer of mathematical education. He made significant contributions to many branches of mathematics, including group theory, representation theory and functional analysis. The recipient of many awards, including the Order of Lenin and the first Wolf Prize, he was a Foreign Fellow of the Royal Society and professor at Moscow State University and, after immigrating to the United States shortly before his 76th birthday, at Rutgers University. Gelfand is also a 1994 MacArthur Fellow. His legacy continues through his students, who include Endre Szemerédi, Alexandre Kirillov, Edward Frenkel, Joseph Bernstein, David Kazhdan, as well as his own son, Sergei Gelfand. Early years A native of Kherson Governorate, Russian Empire (now, ... [...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] |
Coherent Sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X- modules that has a local presentation, that is, every point in X has an open neighborhood U in which there is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Category
In mathematics, especially representation theory, the stable module category is a category in which projectives are "factored out." Definition Let ''R'' be a ring. For two modules ''M'' and ''N'' over ''R'', define \underline(M,N) to be the set of ''R''-linear maps from ''M'' to ''N'' modulo the relation that ''f'' ~ ''g'' if ''f'' − ''g'' factors through a projective module. The stable module category is defined by setting the objects to be the ''R''-modules, and the morphisms are the equivalence classes \underline(M,N). Given a module ''M'', let ''P'' be a projective module with a surjection p \colon P \to M. Then set \Omega(M) to be the kernel of ''p''. Suppose we are given a morphism f \colon M \to N and a surjection q \colon Q \to N where ''Q'' is projective. Then one can lift ''f'' to a map P \to Q which maps \Omega(M) into \Omega(N). This gives a well-defined functor \Omega from the stable module category to itself. For certain rings, such as Fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koszul Algebras
Jean-Louis Koszul (; 3 January 1921 – 12 January 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki. Biography Koszul was educated at the in Strasbourg before studying at the Faculty of Science University of Strasbourg and the Faculty of Science of the University of Paris. His Ph.D. thesis, titled ''Homologie et cohomologie des algèbres de Lie'', was written in 1950 under the direction of Henri Cartan. He lectured at many universities and was appointed in 1963 professor in the Faculty of Science at the University of Grenoble. He was a member of the French Academy of Sciences. Koszul was the cousin of the French composer Henri Dutilleux, and the grandchild of the composer Julien Koszul. Koszul married Denise Reyss-Brion on 17 July 1948. They had three children: Michel, Bertrand, and Anne. He died on 12 January 2018, at the age of 97, nine days after his 97th birthday. See als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X- modules that has a local presentation, that is, every point in X has an open neighborhood U in which there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
