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 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 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.
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.
which is the identity on objects and which sends each morphism to its chain homotopy equivalence class.
The generators and relations construction therefore only guarantees that the morphisms between two objects form a proper class.
For this reason, it is conventional to construct the derived category more concretely even when set theory is not at issue.
The Gabriel–Zisman theorem implies that localization at a multiplicative system has a simple description in terms of roofs.
is a Grothendieck abelian category (meaning that it satisfies AB5 and has a set of generators), with the essential point being that only objects of bounded cardinality are relevant.
[3] In these cases, the limit may be calculated over a small subcategory, and this ensures that the result is a set.
By replacing termwise injectivity by a stronger condition, one gets a similar property that applies even to unbounded complexes.
A theorem of Serpé, generalizing work of Grothendieck and of Spaltenstein, asserts that in a Grothendieck abelian category, every complex is quasi-isomorphic to a K-injective complex with injective terms, and moreover, this is functorial.
As noted before, in the derived category the hom sets are expressed through roofs, or valleys
If one adopts the classical point of view on categories, that there is a set of morphisms from one object to another (not just a class), then one has to give an additional argument to prove this.
If, for example, the abelian category A is small, i.e. has only a set of objects, then this issue will be no problem.
Verdier explained that the definition of the shift X[1] is forced by requiring X[1] to be the cone of the morphism X → 0.
[8] By viewing an object of A as a complex concentrated in degree zero, the derived category D(A) contains A as a full subcategory.
Morphisms in the derived category include information about all Ext groups: for any objects X and Y in A and any integer j, One can easily show that a homotopy equivalence is a quasi-isomorphism, so the second step in the above construction may be omitted.
The definition is usually given in this way because it reveals the existence of a canonical functor In concrete situations, it is very difficult or impossible to handle morphisms in the derived category directly.
Keller also gives applications to Koszul duality, Lie algebra cohomology, and Hochschild homology.
The classical derived functors are related to the total one via RnF(X) = Hn(RF(X)).
For example, the Grothendieck spectral sequence of a composition of two functors such that F maps injective objects in A to G-acyclics (i.e. RiG(F(I)) = 0 for all i > 0 and injective I), is an expression of the following identity of total derived functors J.-L. Verdier showed how derived functors associated with an abelian category A can be viewed as Kan extensions along embeddings of A into suitable derived categories [Mac Lane].