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.