Grothendieck spectral sequence

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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 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)

[edit] 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 ΓY 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(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.

[edit] References


This article incorporates material from Grothendieck spectral sequence on PlanetMath, which is licensed under the GFDL.