and
nonabelian homological algebra
A Grothendieck spectral sequence is a spectral sequence that computes the cochain cohomology of the composite of two derived functors on categories of chain complexes.
Let $\mathcal{A},\mathcal{B},\mathcal{C}$ be abelian categories and let $F \colon \mathcal{A}\to \mathcal{B}$ and $G \colon \mathcal{B}\to \mathcal{C}$ be left exact additive functors. Assume that $\mathcal{A}, \mathcal{B}$ have enough injectives.
Write $R_F\subset \mathrm{Ob} A$ and $R_G\subset\mathrm{Ob} B$ for the classes of objects adapted to $F$ and $G$ respectively, and let furthermore $F(R_A)\subset R_B$. Then the derived functors $R F:D^+(A)\to D^+(B)$, $R G:D^+(B)\to D^+(C)$ and $R(G\circ F):D^+(A)\to D^+(C)$ are defined and the natural morphism $R(G\circ F)\to R G\circ R F$ is an isomorphism.
In the above situation, assume that for every injective object $I \in \mathcal{A}$ the object $F(I) \in \mathcal{B}$ is a $G$-acyclic object.
Then for every object $A \in \mathcal{A}$ there is a spectral sequence $\{E^r_{p,q}(A)\}_{r,p,q}$ called the Grothendieck spectral sequence whose $E_2$-page is the composite
of the right derived functors of $F$ and $G$ in degrees $q$ and $p$, respectively and which is converging to to the derived functors $R^n(G\circ F)$ of the composite of $F$ and $G$:
Moreover, this is natural in $A \in \mathcal{A}$.
By assumption of enough injectives, we may find an injective resolution
of $A$. Next, by the discussion at injective resolution β Existence and construction we may find a fully injective resolution of the chain complex $F(C^\bullet)$:
where hence $I^{\bullet, \bullet}$ is a double complex of injective objects such that for each $n \in \mathbb{N}$ the component $0 \to F(C^n) \to I^{n,\bullet}$ is an ordinary injective resolution of $F(C^n) \in \mathcal{B}$.
Thus we have the corresponding double complex $G(I^{\bullet,\bullet})$ in $\mathcal{C}$. The claim is that the Grothendieck spectral sequence is the spectral sequence of a double complex for $G(I^{\bullet, \bullet})$ equipped with the vertical-degree filtration $\{{}^{vert}E^r_{p,q}(A)\}$:
To see this, notice that by the assumption that $I^{\bullet,\bullet}$ is a fully injective projective resolution, the short exact sequences
are split (by the discussion there) and hence so is their image under any functor and hence in particular under $G$. Accordingly we have
(the first two equivalences by general properties of the filtration spectral sequence, the last by the above splitness). Hence it follows that
where in the last step we used that $H^{q,\bullet}$ is be construction an injective resolution of $H^q(F(C^\bullet)) \simeq R^q F(A)$ (using the $G$-acyclicity of $F(C^\bullet)$).
This establishes the spectral sequence and its second page as claimed. It remains to determine its convergence.
To that end, conside dually, the spectral sequence $\{{}^{hor}E^{p,q}_r\}$ coming from the horizontal filtration on the double complex $G(I^{\bullet, \bullet})$. By the general properties of spectral sequence of a double complex this converges to the same value as the previous one. But for this latter spectral sequence we find
the first equivalence by the general properties of filtration spectral sequences, the second then by the definition of right derived functors. But by assumption $F(C^p)$ is $F$-acyclic and hence all these derived functors vanish in positive degree, so that
Next, the $E_2$-page then contains just horizontal homology of $G(F(C^\bullet))$ and this is by definition now the derived functor of the composite of $F$ with $G$:
Since this is concentrated in the $(q = 0)$-row the spectral sequence of the horizontal filtration collapses here and hence
So in conclusion we have
Many other classes of spectral sequences are special cases of the Grothendieck spectral sequence, for instance the
Leture notes include
James Milne, section 10 of Lectures on Γtale Cohomology
Jinhyun Park, Personal notes on Grothendieck spectral sequence (pdf)