Editing Derived functor (section)

== Variations ==

If one starts with a covariant ''right-exact'' functor <math>G</math>, and the category '''A''' has enough projectives (i.e. for every object <math>A</math> of '''A''' there exists an epimorphism <math>P\rightarrow A</math> where <math>P</math> is a [[projective module|projective object]]), then one can define analogously the left-derived functors <math>L_iG</math>. For an object <math>X</math> of '''A''' we first construct a projective resolution of the form
:<math>\cdots\to P_2\to P_1\to P_0 \to X \to 0</math>
where the <math>P_i</math> are projective. We apply <math>G</math> to this sequence, chop off the last term, and compute homology to get <math>L_iG(X)</math>. As before, <math>L_0G(X)=G(X)</math>.

In this case, the long exact sequence will grow "to the left" rather than to the right:
:<math>0\to A \to B \to C \to 0</math>
is turned into 
:<math>\cdots\to L_2G(C) \to L_1G(A) \to L_1G(B)\to L_1G(C)\to G(A)\to G(B)\to G(C)\to 0</math>.

Left derived functors are zero on all projective objects.

One may also start with a ''contravariant'' left-exact functor <math>F</math>; the resulting right-derived functors are then also contravariant. The short exact sequence

:<math>0\to A \to B \to C \to 0</math>

is turned into the long exact sequence

:<math>0\to F(C)\to F(B)\to F(A)\to R^1F(C) \to R^1F(B) \to R^1F(A)\to R^2F(C)\to \cdots</math>

These left derived functors are zero on projectives and are therefore computed via projective resolutions.