For a continuous map $f : X \to Y$ and a sheaf $\mathcal{F}$ on $X$:

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

This computes sheaf cohomology on $X$ from the cohomology of $Y$ with coefficients in the higher direct images. It arises from the composition $R\Gamma_Y \circ Rf_* = R\Gamma_X$.

For composable left exact functors $G \circ F$ where $F$ sends injectives to $G$-acyclic objects:

$E_2^{p,q} = R^pG(R^qF(A)) \Rightarrow R^{p+q}(G \circ F)(A)$

The Leray spectral sequence is the special case $F = f_*$, $G = \Gamma_Y$.

For a normal subgroup $N \trianglelefteq G$ and a $G$-module $M$:

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

This computes group cohomology of $G$ from the cohomology of $N$ and $G/N$. It arises from the composition of invariant functors $(-)^G = ((-)^N)^{G/N}$.

For spaces $X, Y$ and a sheaf (or coefficient ring $R$):

$E_2^{p,q} = \bigoplus_{i+j=q} \mathrm{Tor}_p^R(H^i(X, R), H^j(Y, R)) \Rightarrow H^{p+q}(X \times Y, R)$

When $R$ is a field, $\mathrm{Tor} = 0$ for $p \geq 1$ and this collapses to the Kunneth formula $H^n(X \times Y) \cong \bigoplus_{i+j=n} H^i(X) \otimes H^j(Y)$.

For a chain complex $C_\bullet$ of free abelian groups and an abelian group $G$:

$E_2^{p,q} = \mathrm{Ext}^p_{\mathbb{Z}}(H_q(C), G) \Rightarrow H^{p+q}(C; G)$

For $\mathbb{Z}$-coefficients this degenerates to the universal coefficient short exact sequence since $\mathrm{Ext}^p = 0$ for $p \geq 2$.

A double complex $K^{p,q}$ with two commuting differentials $d_h$ and $d_v$ has two spectral sequences converging to $H^n(\mathrm{Tot}(K))$. The first has $E_1^{p,q} = H^q_v(K^{p,\bullet})$ and the second has $E_1^{p,q} = H^p_h(K^{\bullet, q})$. Comparing them is a powerful technique.

For a smooth projective variety $X$ over a field:

$E_1^{p,q} = H^q(X, \Omega^p_X) \Rightarrow H^{p+q}_{\mathrm{dR}}(X)$

In characteristic 0, this degenerates at $E_1$ (by Hodge theory), giving the Hodge decomposition $H^n_{\mathrm{dR}}(X) \cong \bigoplus_{p+q=n} H^q(X, \Omega^p)$. In positive characteristic, it can fail to degenerate.

For a generalized cohomology theory $h^\bullet$ and a CW complex $X$:

$E_2^{p,q} = H^p(X; h^q(\mathrm{pt})) \Rightarrow h^{p+q}(X)$

This computes generalized cohomology from ordinary cohomology. For $K$-theory, $h^q(\mathrm{pt}) = \mathbb{Z}$ for $q$ even and $0$ for $q$ odd.

A spectral sequence collapses at $E_2$ if $d_r = 0$ for all $r \geq 2$, so $E_\infty = E_2$. This gives $H^n \cong \bigoplus_{p+q=n} E_2^{p,q}$ (the filtration splits). Collapsing often follows from vanishing of certain groups or from weight/purity arguments.

A spectral sequence $E_2^{p,q} \Rightarrow H^{p+q}$ has edge maps: $H^n \twoheadrightarrow E_\infty^{n,0} \hookrightarrow E_2^{n,0}$ and $E_2^{0,n} \twoheadrightarrow E_\infty^{0,n} \hookrightarrow H^n$. For the Leray spectral sequence, the edge map $H^n(X) \to E_2^{0,n} = H^0(Y, R^nf_*\mathcal{F})$ is the restriction to fibers, and $E_2^{n,0} = H^n(Y, f_*\mathcal{F}) \to H^n(X)$ is the pullback.

Any spectral sequence $E_2^{p,q} \Rightarrow H^{p+q}$ gives a five-term exact sequence:

$0 \to E_2^{1,0} \to H^1 \to E_2^{0,1} \xrightarrow{d_2} E_2^{2,0} \to H^2$

For the Hochschild-Serre sequence: $0 \to H^1(G/N, M^N) \to H^1(G, M) \to H^1(N, M)^{G/N} \to H^2(G/N, M^N) \to H^2(G, M)$, which is the inflation-restriction exact sequence.

In the derived category, a spectral sequence arises whenever one computes the composition $RG \circ RF$ versus $R(G \circ F)$. The spectral sequence is a way of extracting cohomological information from the total derived functor. In principle, the total derived functor $R(G \circ F)$ contains all the information, and the spectral sequence is a computational tool for extracting it.