Definitions

# Grothendieck spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a technique that allows one to compute the derived functors of the composition of two functors $Gcirc F$, from knowledge of the derived functors of F and G.

If

$F :mathcal\left\{C\right\}tomathcal\left\{D\right\}$

and

$G :mathcal\left\{D\right\}tomathcal\left\{E\right\}$

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

$E_2^\left\{pq\right\} = \left(\left\{rm R\right\}^p G circ\left\{rm R\right\}^q F\right)\left(A\right) implies \left\{rm R\right\}^\left\{p+q\right\} \left(Gcirc F\right)\left(A\right)$

Many spectral sequences are merely instances of the Grothendieck spectral sequence, for example the Leray spectral sequence and the Lyndon-Hochschild-Serre 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

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

$f : X to Y$

there is the (left-exact) direct image functor

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

We also have the global section functors

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

and

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

Then since

$Gamma_Y circ f_* = Gamma_X$

and the functors $f_*$ and $Gamma_Y$ satisfy the hypotheses (injectives are flasque sheaves, direct images of flasque sheaves are flasque, and flasque sheaves are acyclic for the global section functor), the sequence in this case becomes:

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

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

## References

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