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In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). The construction proceeds on the basis that the objects of D(C) should be chain complexes from C, identified in a certain way that in a sense absorbs the usual long exact sequences, provided by the snake lemma. There are in fact a few versions, depending on conditions bounding the chain complexes in various ways. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...
Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...
In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ...
The development of the derived category, by Alexander Grothendieck and his student 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 remarkable strides and became close to appearing a universal approach in mathematics. The basic theory of Verdier was written down in his dissertation, never to be published (a summary much later appeared in SGA4½). The axiomatics required an innovation, the concept of triangulated category, as well as localization of a category, and at least one notorious axiom (octahedral axiom). Such was the style of abstraction of the time. In fact there was a pressing concern, to get a neat formulation of coherent duality, that explains how the 'derived' way of thinking was ever launched. (It has later been hailed, for example by Manin, as a rectification of the deficiencies of the established Cartan-Eilenberg method of accepting derived functors such as the Tor functors and Ext functors as natural.) Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. ...
In mathematics, Alexander Grothendiecks Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois...
In mathematics, a triangulated category is a category satisfying some axioms that are based on the properties of a derived 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. ...
Coherent duality in mathematics refers to a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local theory. ...
Yuri Ivanovitch Manin (born 1937) is a Russian-born mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. ...
The Tor functors are the derived functors of the tensor product functor in mathematics. ...
In mathematics, the Ext functors of homological algebra are derived functors of functors. ...
In coherent sheaf theory, pushing to the limit of what could be done with Serre duality without the assumption of a non-singular scheme, the need to take a whole complex of sheaves became apparent. In fact the Cohen-Macaulay ring condition, a weakening of non-singularity, corresponds to the existence of a single dualizing sheaf; and this is far from the general case. From the top-down intellectual position, always assumed by Grothendieck, this signified a need to reformulate. With it came the idea that the 'real' tensor product and Hom functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational devices. In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn. ...
In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. ...
In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ...
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
In mathematics, a Cohen-Macaulay ring is a commutative noetherian local ring with Krull dimension equal to its depth. ...
In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...
Despite the level of abstraction, the derived category methodology established itself over the following decades; and perhaps began to impose itself with the formulation of the Riemann-Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980. The Sato school adopted it, and the subsequent history of D-modules was of a theory expressed in those terms. Mikio Sato (佐藤 幹夫, born April 18, 1928) is a Japanese mathematician, working in what he calls algebraic analysis. ...
A parallel development, speaking in fact the same language, was that of spectrum in homotopy theory. This was at the space level, rather than in the algebra. In mathematics, a spectrum in homotopy theory is an object in a category constructed for the purposes of stable homotopy theory, starting with the category of CW complexes and aiming to make the suspension functor S invertible. ...
An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
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