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 $G\circ F$, 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 (covariant) functors between abelian categories such that G is left exact and 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 $\mathcal{C}$:

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

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 Γ_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.

References

• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR1269324 

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