# Derived functor

A functor "measuring" the deviation of a given functor from being exact. Let $T ( A , C )$ be an additive functor from the product of the category of $R _ {1}$- modules with the category of $R _ {2}$- modules into the category of $R$- modules that is covariant in the first argument and contravariant in the second argument. From an injective resolution $X$ of $A$ and a projective resolution $Y$ of $C$ one obtains a doubly-graded complex $T( X , Y )$. The homology of the associated single complex $T ( A , C )$ does not depend on the choice of resolutions, has functorial properties and is called the right derived functor $R ^ {n} T ( A , C )$ of $T ( A , C )$. The basic property of a derived functor is the existence of long exact sequences

$$\rightarrow R ^ {n} T ( A ^ \prime , C ) \rightarrow R ^ {n} T ( A , C ) \rightarrow R ^ {n} T ( A ^ {\prime\prime} , C ) \rightarrow$$

$$\rightarrow \ R ^ {n+} 1 T ( A ^ \prime , C ) \rightarrow \dots$$

$$\rightarrow R ^ {n} T ( A , C ^ {\prime\prime} ) \rightarrow R ^ {n} T ( A , C ) \rightarrow R ^ {n} T ( A , C ^ \prime ) \rightarrow$$

$$\rightarrow \ R ^ {n+} 1 T ( A , C ^ {\prime\prime} ) \rightarrow \dots ,$$

induced by short exact sequences

$$0 \rightarrow A ^ \prime \rightarrow A \rightarrow A ^ {\prime\prime} \rightarrow 0,$$

$$0 \rightarrow C ^ \prime \rightarrow C \rightarrow C ^ {\prime\prime} \rightarrow 0 .$$

The left derived functor is defined analogously. The derived functor of $\mathop{\rm Hom} _ {R}$ is denoted by $\mathop{\rm Ext} _ {R} ^ {n}$. The group $\mathop{\rm Ext} _ {R} ^ {1} ( A , C )$ classifies extensions of $A$ with kernel $C$ up to equivalence (cf. Baer multiplication; Cohomology of algebras).

How to Cite This Entry:
Derived functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derived_functor&oldid=46635
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article