For a left exact functor $F$, we have $R^0F \cong F$: since $F$ preserves the exactness of $0 \to A \to I^0 \to I^1$, we get $H^0(F(I^\bullet)) = \ker(F(I^0) \to F(I^1)) \cong F(A)$. The higher $R^nF$ measure the failure of $F$ to be exact.

The global sections functor $\Gamma(X, -) : \mathbf{Sh}(X) \to \mathbf{Ab}$ is left exact. Its right derived functors are sheaf cohomology: $H^n(X, \mathcal{F}) = R^n\Gamma(X, \mathcal{F})$. Compute by taking an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ and setting $H^n(X, \mathcal{F}) = H^n(\Gamma(X, \mathcal{I}^\bullet))$.

For a group $G$, the functor $(-)^G : G\text{-}\mathbf{Mod} \to \mathbf{Ab}$ sending $M$ to the $G$-fixed points $M^G = \mathrm{Hom}_{\mathbb{Z}[G]}(\mathbb{Z}, M)$ is left exact. Its right derived functors give group cohomology: $H^n(G, M) = R^n((-)^G)(M) = \mathrm{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z}, M)$.

For a Lie algebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $M$, the functor of $\mathfrak{g}$-invariants $M \mapsto M^{\mathfrak{g}} = \mathrm{Hom}_{U(\mathfrak{g})}(k, M)$ is left exact. Its right derived functors give Lie algebra cohomology: $H^n(\mathfrak{g}, M) = \mathrm{Ext}^n_{U(\mathfrak{g})}(k, M)$.

For a right exact functor $G$, we have $L_0G \cong G$. The classical example: $G = M \otimes_R -$ is right exact, and $L_n(M \otimes_R -)(N) = \mathrm{Tor}_n^R(M, N)$.

$R^nF(I) = 0$ for all $n > 0$ when $I$ is injective (since $I$ is its own injective resolution). More generally, $R^nF(A) = 0$ for $n > 0$ if $A$ is $F$-acyclic (i.e., $R^nF(A) = 0$ for $n \geq 1$). Acyclic objects can be used in place of injectives for computing derived functors.

For a good cover $\mathcal{U}$ of $X$, Cech cohomology $\check{H}^n(\mathcal{U}, \mathcal{F})$ agrees with the derived functor cohomology $H^n(X, \mathcal{F})$ when $\mathcal{F}$ is a sheaf on a paracompact Hausdorff space or when the cover is acyclic for $\mathcal{F}$.

On a smooth manifold $X$, the de Rham complex $\Omega^\bullet_X$ is a resolution of the constant sheaf $\mathbb{R}_X$ (by the Poincare lemma). Hence $H^n_{\mathrm{dR}}(X) = H^n(\Gamma(X, \Omega^\bullet)) \cong H^n(X, \mathbb{R}_X) = R^n\Gamma(X, \mathbb{R}_X)$.

A short exact sequence $0 \to A \to B \to C \to 0$ in $\mathcal{A}$ induces a long exact sequence

$0 \to F(A) \to F(B) \to F(C) \to R^1F(A) \to R^1F(B) \to R^1F(C) \to R^2F(A) \to \cdots$

This is the fundamental computational tool: knowing $R^nF$ on two of three terms determines the third (up to extensions).

For a morphism $f : X \to Y$ of topological spaces (or schemes), the direct image $f_* : \mathbf{Sh}(X) \to \mathbf{Sh}(Y)$ is left exact. Its right derived functors $R^nf_*$ are the higher direct image sheaves. The stalk $(R^nf_*\mathcal{F})_y \cong H^n(f^{-1}(y), \mathcal{F}|_{f^{-1}(y)})$ (under mild hypotheses).

The cohomological dimension of an object $A$ with respect to $F$ is $\mathrm{cd}_F(A) = \sup\{n : R^nF(A) \neq 0\}$. For sheaf cohomology on an $n$-dimensional manifold, Grothendieck's vanishing theorem gives $H^k(X, \mathcal{F}) = 0$ for $k > n$.

In the derived category, the total right derived functor $RF : D^+(\mathcal{A}) \to D^+(\mathcal{B})$ is defined by $RF(A^\bullet) = F(I^\bullet)$ where $A^\bullet \to I^\bullet$ is an injective resolution in $D^+$. The classical derived functors are recovered as $R^nF(A) = H^n(RF(A[0]))$.