# Derived functor

A functor "measuring" the deviation of a given functor from being exact. Let be an additive functor from the product of the category of -modules with the category of -modules into the category of -modules that is covariant in the first argument and contravariant in the second argument. From an injective resolution of and a projective resolution of one obtains a doubly-graded complex . The homology of the associated single complex does not depend on the choice of resolutions, has functorial properties and is called the right derived functor of . The basic property of a derived functor is the existence of long exact sequences

induced by short exact sequences

The left derived functor is defined analogously. The derived functor of is denoted by . The group classifies extensions of with kernel up to equivalence (cf. Baer multiplication; Cohomology of algebras).

#### References

[1] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) |

[2] | S. MacLane, "Homology" , Springer (1963) |

#### Comments

The above article does not explain the sense in which measures the deviation of from being exact. The point is that if is left exact (i.e. preserves the exactness of sequences of the form in the fist variable and of the form in the second), then is naturally isomorphic to ; if further is exact, then for all . Derived functors may also be defined for additive functors of a single variable between module categories, or, more generally, between arbitrary Abelian categories, provided the necessary injective or projective resolutions exist in the domain category.

**How to Cite This Entry:**

Derived functor.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Derived_functor&oldid=15566