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Derived Noncommutative Algebraic Geometry
In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, D^b(X), called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted D_(X). For instance, the derived category of coherent sheaves D^b(X) on a smooth projective variety can be used as an invariant of the underlying variety for many cases (if X has an ample (anti-)canonical sheaf). Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name. Derived category of projective line The derived category of \mathbb^1 is one of the motivating examples for derived non-commutative schemes due to its easy categorical structure. Recall that the Euler sequence of \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative Algebraic Geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme. Functions on usual spaces in the traditional (commutative) algebraic geometry have a product defined by pointwise multiplication; as the values of these functions commute, the functions also commute: '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Category
In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category. Introduction and motivation A category ''C'' consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replace ''C'' by another category ''C in which certain morphisms are forced to be isomorphisms. This process is called localization. For example, in the category of ''R''-modules (for some fixed commutative ring ''R'') the multiplication by a fixed element ''r'' of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1201
Year 1201 ( MCCI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. Events By place Byzantine Empire * July 31 – John Komnenos the Fat, a Byzantine aristocrat, attempts to usurp the imperial throne; he is proclaimed emperor and crowned by Patriarch John X Kamateros, at Constantinople. Meanwhile, Emperor Alexios III Angelos, who resides in the Palace of Blachernae, dispatches a small force under Alexios Palaiologos, Alexios' son-in-law, who is regarded as his heir-apparent. With support of the Varangian Guard, John is overthrown and decapitated by the end of the day. His head is displayed at the Forum of Constantine, while John's supporters are captured and tortured to extract the names of all the conspirators. * Autumn – Prince Alexios Angelos, son of the deposed, blinded and imprisoned late Emperor Isaac II Angelos, escapes from Constantinople. He makes his way to Sicily and then Rome where he is turned a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Mirror Symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address to the 1994 International Congress of Mathematicians in Zürich, speculated that mirror symmetry for a pair of Calabi–Yau manifolds ''X'' and ''Y'' could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of ''X'' (the derived category of coherent sheaves on ''X'') and another triangulated category constructed from the symplectic geometry of ''Y'' (the derived Fukaya category). Edward Witten originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B model topological string theories. These models concern maps from Riemann surfaces into a fixed target—usually a Calabi–Yau manifold. Most of the mathematical predictions of mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bridgeland Stability Condition
In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category. The case of original interest and particular importance is when this derived category is the derived category of coherent sheaves on a Calabi–Yau manifold, and this situation has fundamental links to string theory and the study of D-branes. Such stability conditions were introduced in a rudimentary form by Michael Douglas Michael Kirk Douglas (born September 25, 1944) is an American actor and film producer. He has received numerous accolades, including two Academy Awards, five Golden Globe Awards, a Primetime Emmy Award, the Cecil B. DeMille Award, and the A ... called \Pi-stability and used to study BPS B-branes in string theory.Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006. This concep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiorthogonal Decomposition
In mathematics, a semiorthogonal decomposition is a way to divide a triangulated category into simpler pieces. One way to produce a semiorthogonal decomposition is from an exceptional collection, a special sequence of objects in a triangulated category. For an algebraic variety ''X'', it has been fruitful to study semiorthogonal decompositions of the bounded derived category of coherent sheaves, \text^(X). Semiorthogonal decomposition Alexei Bondal and Mikhail Kapranov (1989) defined a semiorthogonal decomposition of a triangulated category \mathcal to be a sequence \mathcal_1,\ldots,\mathcal_n of strictly full triangulated subcategories such that: *for all 1\leq i [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Complex
In algebra, a perfect complex of modules over a commutative ring ''A'' is an object in the derived category of ''A''-modules that is quasi-isomorphic to a bounded complex of finite projective ''A''-modules. A perfect module is a module that is perfect when it is viewed as a complex concentrated at degree zero. For example, if ''A'' is Noetherian, a module over ''A'' is perfect if and only if it is finitely generated and of finite projective dimension. Other characterizations Perfect complexes are precisely the compact objects in the unbounded derived category D(A) of ''A''-modules. They are also precisely the dualizable objects in this category. A compact object in the ∞-category of (say right) module spectra over a ring spectrum is often called perfect;http://www.math.harvard.edu/~lurie/281notes/Lecture19-Rings.pdf see also module spectrum. Pseudo-coherent sheaf When the structure sheaf \mathcal_X is not coherent, working with coherent sheaves has awkwardness (namely the k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangulated Category
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 conj ... [...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 made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bertini's Theorem
In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0. Statement for hyperplane sections of smooth varieties Let ''X'' be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space \mathbf P^n. Let , H, denote the complete system of hyperplane divisors in \mathbf P^n. Recall that it is the dual space (\mathbf P^n)^ of \mathbf P^n and is isomorphic to \mathbf P^n. The theorem of Bertini states that the set of hyperplanes not containing ''X'' and with smooth intersection with ''X'' contains an open dense subset of the total system of divisors , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flat Module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Flatness was introduced by in his paper '' Géometrie Algébrique et Géométrie Analytique''. See also flat morphism. Definition A module over a ring is ''flat'' if the following condition is satisfied: for every injective linear map \varphi: K \to L of -modules, the map :\varphi \otimes_R M: K \otimes_R M \to L \otimes_R M is also injective, where \varphi \otimes_R M is the map induced by k \otimes m \mapsto \varphi(k) \otimes m. For this definition, it is enough to restrict the injections \varphi to the inclusions of finitely generated ideals into . Equivalently, an -module is flat if the tensor product with is an exact functor; that is if, for every short exact sequence of - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
