Grothendieck spectral sequence

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{C}tomathcal{D}

and

G :mathcal{D}tomathcal{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} (Gcirc 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)

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 (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.

References


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