= Right derived functor
{title2=$R^iF$}
For an additive covariant <left-exact functor> on module categories, choose an <injective resolution> $M\to I^\bullet$ and set $R^iF(M)=H^i(F(I^\bullet))$. For a contravariant functor, use a <projective resolution> instead, as in <contravariant right derived functor>. Resolution comparison and homotopies give independence and functoriality; short exact sequences give long exact derived-functor sequences. The zeroth functor is naturally $F$.
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