# Definition:Right Derived Functor

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## Definition

Let $\mathbf A$ be an abelian category with enough injectives.

Let $\mathbf B$ be an abelian category.

Let $F: \mathbf A \to \mathbf B$ be a left exact functor.

Let $X$ and $Y$ be objects of $\mathbf A$.

Let $f: X \to Y$ be a morphism of $\mathbf A$.

Let $I$ be an arbitrary injective resolution of $X$.

Let $J$ be an arbitrary injective resolution of $Y$.

Let $\tilde f : I \to J$ be a morphism of cochain complexes induced by $f$.

Let $\map F I$ denote the cochain complex defined by applying the functor on cochains induced by $F$ to $I$.

Let $i \in \Z_{\ge 0}$ be a non-negative integer.

Let $\map {H^i} {\map F I}$ denote the $i$-th cohomology of $\map F I$.

The $i$-th right derived functor $\mathrm R^i F : \mathbf A \to \mathbf B$ of $F$ is defined on objects as:

$\mathrm R^i \map F X := \map {H^i} {\map F I}$

The $i$-th right derived functor $\mathrm R^i F$ of $F$ is defined on morphisms as follows:

Define $\mathrm R^i \map F f: \mathrm R^i \map F X \to \mathrm R^i \map F Y$ by the induced map $\map {H^i} {\map F {\tilde f} } : \map {H^i} {\map F I} \to \map {H^i} {\map F J}$.