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derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
== Motivation ==
It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations.
Suppose we are given a covariant left exact functor ''F'' : A → B between two abelian categories A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) there is one canonical way of doing so, given by the right derived functors of ''F''. For every ''i''≥1, there is a functor ''RiF'': A → B, and the above sequence continues like so: 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') → ''R''1''F''(''A'') → ''R''1''F''(''B'') → ''R''1''F''(''C'') → ''R''2''F''(''A'') → ''R''2''F''(''B'') → ... . From this we see that ''F'' is an exact functor if and only if ''R''1''F'' = 0; so in a sense the right derived functors of ''F'' measure "how far" ''F'' is from being exact.
If the object ''A'' in the above short exact sequence is injective, then the sequence splits. Applying any additive functor to a split sequence results in a split sequence, so in particular ''R''1''F''(''A'') = 0. Right derived functors (for ''i>0'') are zero on injectives: this is the motivation for the construction given below.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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