Editing Homological algebra (section)

===Derived functor===
{{Main|Derived functor}}
Suppose we are given a covariant [[left exact functor]] ''F'' : '''A''' → '''B''' between two [[abelian category|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 form|canonical]] way of doing so, given by the right derived functors of ''F''. For every ''i''≥1, there is a functor ''R<sup>i</sup>F'': '''A''' → '''B''', and the above sequence continues like so: 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') → ''R''<sup>1</sup>''F''(''A'') → ''R''<sup>1</sup>''F''(''B'') → ''R''<sup>1</sup>''F''(''C'') → ''R''<sup>2</sup>''F''(''A'') → ''R''<sup>2</sup>''F''(''B'') → ... . From this we see that ''F'' is an exact functor if and only if ''R''<sup>1</sup>''F'' = 0; so in a sense the right derived functors of ''F'' measure "how far" ''F'' is from being exact.