Take $F = f_* : \mathbf{Sh}(X) \to \mathbf{Sh}(Y)$ and $G = \Gamma(Y, -) : \mathbf{Sh}(Y) \to \mathbf{Ab}$. Then $G \circ F = \Gamma(X, -)$ and $R^qF = R^qf_*$, $R^pG = H^p(Y, -)$. The Grothendieck spectral sequence gives:

$E_2^{p,q} = H^p(Y, R^qf_*\mathcal{F}) \Rightarrow H^{p+q}(X, \mathcal{F})$

The acyclicity condition holds because $f_*$ preserves injective sheaves (as it has an exact left adjoint $f^{-1}$).

For $N \trianglelefteq G$, take $F = (-)^N : G\text{-}\mathbf{Mod} \to (G/N)\text{-}\mathbf{Mod}$ and $G = (-)^{G/N}$. Then $G \circ F = (-)^G$ and:

$E_2^{p,q} = H^p(G/N, H^q(N, M)) \Rightarrow H^{p+q}(G, M)$

The acyclicity condition holds: $(-)^N$ sends injective $G$-modules to injective $G/N$-modules.

For sheaves $\mathcal{E}, \mathcal{F}$ on a scheme $X$, take $F = \mathcal{H}om(\mathcal{E}, -)$ (the sheaf Hom) and $G = \Gamma(X, -)$. Then:

$E_2^{p,q} = H^p(X, \mathcal{E}xt^q(\mathcal{E}, \mathcal{F})) \Rightarrow \mathrm{Ext}^{p+q}(\mathcal{E}, \mathcal{F})$

This relates the local Ext sheaves to the global Ext groups.

For a group extension $1 \to N \to G \to Q \to 1$ (not necessarily split), the spectral sequence $H^p(Q, H^q(N, M)) \Rightarrow H^{p+q}(G, M)$ reduces to the five-term exact sequence:

$0 \to H^1(Q, M^N) \xrightarrow{\mathrm{inf}} H^1(G, M) \xrightarrow{\mathrm{res}} H^1(N, M)^Q \xrightarrow{d_2} H^2(Q, M^N) \xrightarrow{\mathrm{inf}} H^2(G, M)$

For an open cover $\mathcal{U}$ of $X$, there is a spectral sequence $\check{H}^p(\mathcal{U}, \mathcal{H}^q(\mathcal{F})) \Rightarrow H^{p+q}(X, \mathcal{F})$ where $\mathcal{H}^q$ denotes the presheaf $U \mapsto H^q(U, \mathcal{F})$. For a Leray cover (where all intersections are acyclic), $\mathcal{H}^q = 0$ for $q > 0$ and the spectral sequence degenerates.

For maps $X \xrightarrow{f} Y \xrightarrow{g} Z$:

$E_2^{p,q} = R^pg_*(R^qf_*\mathcal{F}) \Rightarrow R^{p+q}(g \circ f)_*\mathcal{F}$

This computes the higher direct images of the composite from those of $f$ and $g$ individually.

The Grothendieck spectral sequence degenerates at $E_2$ (i.e., $E_2 = E_\infty$) when $R^qF(A) = 0$ for $q \neq 0$ (so the aisle is concentrated on one line) or when $R^pG = 0$ for $p \neq 0$. In the first case, $R^n(G \circ F)(A) \cong R^nG(F(A))$. In the second, $R^n(G \circ F)(A) \cong G(R^nF(A))$.

The edge maps are: $R^n(G \circ F)(A) \to R^nG(F(A))$ (the natural map from composite derived functors to $G$ derived at $F(A)$) and $G(R^nF(A)) \to R^n(G \circ F)(A)$. These are isomorphisms when the spectral sequence degenerates along the corresponding edge.

The hypothesis that $F$ sends injectives to $G$-acyclics is essential. If $F$ does not preserve acyclicity, the spectral sequence still exists but with a more complicated $E_2$ page involving hyper-derived functors. In practice, the acyclicity condition is almost always satisfied in geometric situations.

In derived categories: $R(G \circ F) \simeq RG \circ RF$ when $F$ preserves $G$-acyclics. The spectral sequence arises from the t-structure filtration on $RG(RF(A))$. The $E_2$ page $R^pG(R^qF(A))$ is the cohomology of the cohomology, and the spectral sequence converges to the cohomology of the total complex.

For $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ where $F$ does NOT send injectives to $G$-acyclics, one can still construct a spectral sequence using Cartan-Eilenberg resolutions (double injective resolutions). The resulting spectral sequence has the same $E_2$ page but the convergence proof is more delicate.

There is a dual version for left derived functors. For right exact functors $F, G$ with $F$ sending projectives to $G$-acyclics:

$E^2_{p,q} = L_pG(L_qF(A)) \Rightarrow L_{p+q}(G \circ F)(A)$

This gives the Kunneth spectral sequence and the universal coefficient spectral sequence as special cases.