Variations
If one starts with a covariant right-exact functor G, and the category A has enough projectives (i.e. for every object A of A there exists an epimorphism P → A where P is a projective object), then one can define analogously the left-derived functors LiG. For an object X of A we first construct a projective resolution of the form
where the Pi are projective. We apply G to this sequence, chop off the last term, and compute homology to get LiG(X). As before, L0G(X) = G(X).
In this case, the long exact sequence will grow "to the left" rather than to the right:
is turned into
- .
Left derived functors are zero on all projective objects.
One may also start with a contravariant left-exact functor F; the resulting right-derived functors are then also contravariant. The short exact sequence
is turned into the long exact sequence
These right derived functors are zero on projectives and are therefore computed via projective resolutions.
Read more about this topic: Derived Functor
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