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).

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 $R ^ {n} T$ measures the deviation of $T$ from being exact. The point is that if $T$ is left exact (i.e. preserves the exactness of sequences of the form $0 \rightarrow A ^ \prime \rightarrow A \rightarrow A ^ {\prime\prime}$ in the fist variable and of the form $C ^ \prime \rightarrow C \rightarrow C ^ {\prime\prime} \rightarrow 0$ in the second), then $R ^ {0} T$ is naturally isomorphic to $T$; if further $T$ is exact, then $R ^ {n} T = 0$ for all $n > 0$. 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.