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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. 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 remarkable strides. The basic theory of Verdier was written down in his dissertation, published finally in 1996 in Astérisque (a summary had earlier appeared in SGA 4½). The axiomatics required an innovation, the concept of triangulated category, and the construction is based on localization of a category, a generalization of localization of a ring. The original impulse to develop the 'derived' formalism came from the need to find a suitable formulation of Grothendieck's coherent duality theory. Derived categories have since become indispensable also outside of algebraic geometry, for example in the formulation of the theory of D-modules and microlocal analysis. Recently derived categories have also become important in areas nearer to physics, such as D-branes and mirror symmetry. 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 in place of a single dualizing sheaf 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, derived categories became accepted over the following decades, especially as a convenient setting for sheaf cohomology. Perhaps the biggest advance was the formulation of the Riemann–Hilbert correspondence in dimensions greater than 1 in derived terms, around 1980. The Sato school adopted the language of derived categories, and the subsequent history of D-modules was of a theory expressed in those terms. A parallel development was the category of spectra in homotopy theory. The homotopy category of spectra and the derived category of a ring are both examples of triangulated categories. Let A be an abelian category. (Some basic examples are the category of modules over a ring, or the category of sheaves of abelian groups on a topological space.) We obtain the derived category D(A) in several steps: The second step may be bypassed since a homotopy equivalence is in particular a quasi-isomorphism. But then the simple roof definition of morphisms must be replaced by a more complicated one using finite strings of morphisms (technically, it is no longer a calculus of fractions). So the one-step construction is more efficient in a way, but more complicated. From the point of view of model categories, the derived category D(A) is the true 'homotopy category' of the category of complexes, whereas K(A) might be called the 'naive homotopy category'. For certain purposes (see below) one uses bounded-below ( X n = 0 {displaystyle X^{n}=0} for n ≪ 0 {displaystyle nll 0} ), bounded-above ( X n = 0 {displaystyle X^{n}=0} for n ≫ 0 {displaystyle ngg 0} ) or bounded ( X n = 0 {displaystyle X^{n}=0} for | n | ≫ 0 {displaystyle |n|gg 0} ) complexes instead of unbounded ones. The corresponding derived categories are usually denoted D+(A), D−(A) and Db(A), respectively.

[ "Functor", "Limit (category theory)", "Stable module category", "Differential graded category", "Triangulated category", "Category of groups" ]
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