= Left derived functors on an abelian category
{title2=$L_nF(A)=H_n(FP_\bullet)$}
For a <right-exact additive functor> out of an <abelian category with enough projectives>, apply the <functor> to a <projective resolution> and take <homology objects>. The zeroth derived <functor> is $F$, positive derived <functors> vanish on projective objects, and the degreewise split <horseshoe lemma> produces a long exact sequence from every <short exact sequence in an abelian category>. The target can be any abelian <category>, including an <opposite category> of one.
Back to article page
