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Matrix Factorizations (algebra)
In homological algebra, a branch of mathematics, a matrix factorization is a tool used to study infinitely long resolutions, generally over commutative rings. Motivation One of the problems with non-smooth algebras, such as Artin algebras, are their derived categories are poorly behaved due to infinite projective resolutions. For example, in the ring R = \mathbb (x^2) there is an infinite resolution of the R-module \mathbb where\cdots \xrightarrow R \xrightarrow R \xrightarrow R \to \mathbb \to 0Instead of looking at only the derived category of the module category, David Eisenbud studied such resolutions by looking at their periodicity. In general, such resolutions are periodic with period 2 after finitely many objects in the resolution. Definition For a commutative ring S and an element f \in S, a matrix factorization of f is a pair of n\times n square matrices A,B such that AB = f \cdot \text_n. This can be encoded more generally as a \mathbb/2 graded S-module M = M_0\oplus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has played a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolution (algebra)
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero-object. Generally, the objects in the sequence are restricted to have some property ''P'' (for example to be free). Thus one speaks of a ''P resolution''. In particular, every module has free resolutions, projective reso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Algebra
In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring ''R'' that is a finitely generated ''R''-module. They are named after Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar .... Every Artin algebra is an Artin ring. Dual and transpose There are several different dualities taking finitely generated modules over Λ to modules over the opposite algebra Λop. *If ''M'' is a left Λ module then the right Λ-module ''M''* is defined to be HomΛ(''M'',Λ). * The dual ''D''(''M'') of a left Λ-module ''M'' is the right Λ-module ''D''(''M'') = Hom''R''(''M'',''J''), where ''J'' is the dualizing module of ''R'', equal to the sum of the injective envelopes of the non-isomorphic simple ''R''-modules or equivalently the injective envelope of ''R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and Director of the Mathematical Sciences Research Institute (MSRI); he previously served as Director of MSRI from 1997 to 2007. Biography Eisenbud is the son of mathematical physicist Leonard Eisenbud, who was a student and collaborator of the renowned physicist Eugene Wigner. Eisenbud received his Ph.D. in 1970 from the University of Chicago, where he was a student of Saunders Mac Lane and, unofficially, James Christopher Robson. He then taught at Brandeis University from 1970 to 1997, during which time he had visiting positions at Harvard University, Institut des Hautes Études Scientifiques (IHÉS), University of Bonn, and Centre national de la recherche scientifique (CNRS). He joined the staff at MSRI in 1997, and took a position at Berkeley at the same time. From 2003 to 2005 Eisenbud was President of the Americ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 In mathe ... [...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 re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. It has played a ... [...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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
