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

:$F\; :mathcal\{C\}\; omathcal\{D\}$

and

:$G\; :mathcal\{D\}\; omathcal\{E\}$

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 $mathcal\{C\}$:

:$E\_2^\{pq\}\; =\; (\{\; m\; R\}^p\; G\; circ\{\; m\; R\}^q\; F)(A)\; implies\; \{\; m\; R\}^\{p+q\}\; (Gcirc\; F)(A)$

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 → "R"¹"G"("FA") → "R"¹("GF")("A") → "G"("R"¹"F"("A")) → "R"²"G"("FA") → "R"²("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

:$f\; :\; X\; o\; Y$

there is the (left-exact) direct image functor

:$f\_*\; :\; mathbf\{Ab\}(X)\; o\; mathbf\{Ab\}(Y)$.

We also have the global section functors

:$Gamma\_X\; :\; mathbf\{Ab\}(X)\; o\; mathbf\{Ab\}$,

and

:$Gamma\_Y\; :\; mathbf\{Ab\}(Y)\; o\; mathbf\; \{Ab\}.$

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:

:$H^p(Y,\{\; m\; R\}^q\; f\_*mathcal\{F\})implies\; H^\{p+q\}(X,mathcal\{F\})$

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

References

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