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Use this tag for questions about a particular construction of homological algebra of an abelian category A that refines and in a certain sense simplifies the theory of derived functors defined on A.

The derived category D(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense simplify the theory of derived functors defined on A. The construction proceeds on the basis that 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. Derived categories are also useful outside of algebraic geometry, for example in the formulation of the theory of D-modules and microlocal analysis and, nearer to physics, D-branes and mirror symmetry.

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 steps:

  • The basic object is the category Kom(A) of chain complexes $$\cdots \to X^{-1} \xrightarrow {d^{-1}} X^0 \xrightarrow {d^0} X^1\xrightarrow {d^1} X^2 \to \cdots$$ in A. Its objects will be the objects of the derived category but its morphisms will be altered.
  • Pass to the homotopy category of chain complexes K(A) by identifying morphisms which are chain homotopic.
  • Pass to the derived category D(A) by localizing at the set of quasi-isomorphisms. Morphisms in the derived category may be explicitly described as roofs XX' → Y, where X' → X is a quasi-isomorphism and X' → Y is any morphism of chain complexes.

The second step may be bypassed because 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."

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