Editing Derived functor (section)

== Construction and first properties ==

The crucial assumption we need to make about our abelian category '''A''' is that it has ''enough injectives'', meaning that for every object ''A'' in '''A''' there exists a [[monomorphism]] ''A'' → ''I'' where ''I'' is an [[injective object]] in '''A'''.

The right derived functors of the covariant left-exact functor ''F'' : '''A''' → '''B''' are then defined as follows. Start with an object ''X'' of '''A'''. Because there are enough injectives, we can construct a long exact sequence of the form
:<math>0\to X\to I^0\to I^1\to I^2\to\cdots</math>
where the ''I''<sup>&nbsp;''i''</sup> are all injective (this is known as an ''[[injective resolution]]'' of ''X''). Applying the functor ''F'' to this sequence, and chopping off the first term, we obtain the [[cochain complex]]

:<math>0\to F(I^0)\to F(I^1) \to F(I^2) \to\cdots</math>

Note: this is in general ''not'' an exact sequence anymore. But we can compute its [[cohomology]] at the ''i''-th spot (the kernel of the map from ''F''(''I''<sup>''i''</sup>) modulo the image of the map to ''F''(''I''<sup>''i''</sup>)); we call the result ''R<sup>i</sup>F''(''X''). Of course, various things have to be checked: the result does not depend on the given injective resolution of ''X'', and any morphism ''X'' → ''Y'' naturally yields a morphism ''R<sup>i</sup>F''(''X'') → ''R<sup>i</sup>F''(''Y''), so that we indeed obtain a functor. Note that left exactness means that
0 → ''F''(''X'') → ''F''(''I''<sup>0</sup>) → ''F''(''I''<sup>1</sup>)
is exact, so ''R''<sup>0</sup>''F''(''X'') = ''F''(''X''), so we only get something interesting for ''i''>0.

(Technically, to produce well-defined derivatives of ''F'', we would have to fix an injective resolution for every object of '''A'''. This choice of injective resolutions then yields functors ''R<sup>i</sup>F''. Different choices of resolutions yield [[naturally isomorphic]] functors, so in the end the choice doesn't really matter.)

The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the [[snake lemma]]. This tells us that the collection of derived functors is a [[Delta-functor|δ-functor]].

If ''X'' is itself injective, then we can choose the injective resolution 0 → ''X'' → ''X'' → 0, and we obtain that ''R<sup>i</sup>F''(''X'') = 0 for all ''i'' ≥ 1. In practice, this fact, together with the long exact sequence property, is often used to compute the values of right derived functors.

An equivalent way to compute ''R<sup>i</sup>F''(''X'') is the following: take an injective resolution of ''X'' as above, and let ''K''<sup>''i''</sup> be the image of the map ''I''<sup>''i''-1</sup>→''I<sup>i</sup>'' (for ''i''=0, define ''I''<sup>''i''-1</sup>=0), which is the same as the kernel of ''I''<sup>''i''</sup>→''I''<sup>''i''+1</sup>. Let φ<sub>''i''</sub>&nbsp;:&nbsp;''I''<sup>''i''-1</sup>→''K''<sup>''i''</sup> be the corresponding surjective map. Then ''R<sup>i</sup>F''(''X'') is the cokernel of ''F''(φ<sub>''i''</sub>).