In mathematics, the derived categoryD(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.
Unbounded derived categories were introduced by Spaltenstein in 1988.
In mathematics, the derivedcategory D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense...
category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category...
derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derivedcategories and...
abelian category C can be viewed as sets of morphisms in a category associated to C, the derivedcategory D(C). The objects of the derivedcategory are complexes...
the right derived functors of R0. The more modern (and more general) approach to derived functors uses the language of derivedcategories. In 1968 Quillen...
SI derived units are units of measurement derived from the seven SI base units specified by the International System of Units (SI). They can be expressed...
the category of modules.) Category of rings Derivedcategory Module spectrum Category of graded vector spaces Category of abelian groups Category of representations...
{}_{A}{\mathsf {M}}} are the categories of right A-modules and left A-modules and D refers to the homotopy category (i.e., derivedcategory). By definition, it...
the category of chain complexes Kom(A) of A and the derivedcategory D(A) of A when A is abelian; unlike the former it is a triangulated category, and...
or volume-minimizing. The category having these branes as its objects is called the Fukaya category. The derivedcategory of coherent sheaves is constructed...
instance, to compute the derived functors of a composition of two functors. Spectral sequences are less essential in the derivedcategory approach, but still...
relating them. These abstract from the category of topological spaces or of chain complexes (derivedcategory theory). The concept was introduced by Daniel...
{\displaystyle G^{(2)},G^{(3)},\ldots } are called the second derived subgroup, third derived subgroup, and so forth, and the descending normal series ⋯...
∞-category of "derived rings." In classical algebraic geometry, the derivedcategory of quasi-coherent sheaves is viewed as a triangulated category, but...
downgraded Iota to a Category 4 hurricane as a result of post season reanalysis, which suggested that there was a high bias in windspeeds derived from the Stepped...
derive from the root word happy. It is differentiated from inflection, which is the modification of a word to form different grammatical categories without...
(cotangent complex), Michel Raynaud, Jean-Louis Verdier (co-founder of the derivedcategory theory), and Pierre Deligne. Collaborators on the SGA projects also...
at a time. In fact, similar arguments as above yield a map in the derivedcategory of sheaves on S': g ∗ R f ∗ ( F ) → R f ∗ ′ ( g ′ ∗ F ) {\displaystyle...
weaker, reconstruction theorems from the derivedcategories of (quasi)coherent sheaves motivating the derived noncommutative algebraic geometry (see just...
Iwasawa theory. The derivedcategory of an abelian category is much used in homological algebra. It is the localization of the category of chain complexes...
The derivedcategory of mixed Hodge modules D b MHM ( X ) {\displaystyle D^{b}{\textbf {MHM}}(X)} is intimately related to the derivedcategory of constructuctible...
functors. Usually these are functors between derivedcategories and so are actually left and right derived functors. the direct image f ∗ {\displaystyle...
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the...
derivedcategory D(A) of a Grothendieck abelian category A admits a unique dg enhancement. Differential algebra Graded (mathematics) Graded category Derivator...
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