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Derived CategoryIn mathematics, the derived category D(C) of a category C is a (highly) abstract 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. 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.) 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. 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. 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.
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