# Grothendieck spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a spectral sequence that computes the derived functors of the composition of two functors $G\circ F$, from knowledge of the derived functors of F and G.

If

$F :\mathcal{C}\to\mathcal{D}$

and

$G :\mathcal{D}\to\mathcal{E}$

are two additive and left exact(covariant) functors between abelian categories such that $F$ takes injective objects of $\mathcal{C}$ to $G$-acyclic objects of $\mathcal{D}$, then there is a spectral sequence for each object $A$ of $\mathcal{C}$:

$E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \Longrightarrow {\rm R}^{p+q} (G\circ F)(A).$

Many spectral sequences are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

The exact sequence of low degrees reads

0 → R1G(FA) → R1(GF)(A) → G(R1F(A)) → R2G(FA) → R2(GF)(A).

## Example: the Leray spectral sequence

If $X$ and $Y$ are topological spaces, let

$\mathcal{C} = \mathbf{Ab}(X)$ and $\mathcal{D} = \mathbf{Ab}(Y)$ be the category of sheaves of abelian groups on X and Y, respectively and
$\mathcal{E} = \mathbf{Ab}$ be the category of abelian groups.

For a continuous map

$f : X \to Y$

there is the (left-exact) direct image functor

$f_* : \mathbf{Ab}(X) \to \mathbf{Ab}(Y)$.

We also have the global section functors

$\Gamma_X : \mathbf{Ab}(X)\to \mathbf{Ab}$,

and

$\Gamma_Y : \mathbf{Ab}(Y) \to \mathbf {Ab}.$

Then since

$\Gamma_Y \circ f_* = \Gamma_X$

and the functors $f_*$ and $\Gamma_Y$ satisfy the hypotheses (since the direct image functor has an exact left adjoint $f^{-1}$, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:

$H^p(Y,{\rm R}^q f_*\mathcal{F})\implies H^{p+q}(X,\mathcal{F})$

for a sheaf $\mathcal{F}$ of abelian groups on $X$, and this is exactly the Leray spectral sequence.