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) |
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
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
