Grothendieck spectral sequence

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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[edit]

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.

References[edit]

This article incorporates material from Grothendieck spectral sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.