# nLab Grothendieck spectral sequence

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

A Grothendieck spectral sequence is a spectral sequence that computes the cochain cohomology of the composite of two derived functors on categories of chain complexes.

## Statement

Let $\mathrm{\pi },\mathrm{\beta ¬},\mathrm{\pi }$ be abelian categories and let $F:\mathrm{\pi }\beta \mathrm{\beta ¬}$ and $G:\mathrm{\beta ¬}\beta \mathrm{\pi }$ be left exact additive functors. Assume that $\mathrm{\pi },\mathrm{\beta ¬}$ have enough injectives.

###### Theorem (Tohoku)

Write ${R}_{F}\beta \mathrm{Ob}A$ and ${R}_{G}\beta \mathrm{Ob}B$ for the classes of objects adapted to $F$ and $G$ respectively, and let furthermore $F\left({R}_{A}\right)\beta {R}_{B}$. Then the derived functors $RF:{D}^{+}\left(A\right)\beta {D}^{+}\left(B\right)$, $RG:{D}^{+}\left(B\right)\beta {D}^{+}\left(C\right)$ and $R\left(G\beta F\right):{D}^{+}\left(A\right)\beta {D}^{+}\left(C\right)$ are defined and the natural morphism $R\left(G\beta F\right)\beta RG\beta RF$ is an isomorphism.

###### Theorem (Tohoku)

In the above situation, assume that for every injective object $I\beta \mathrm{\pi }$ the object $F\left(I\right)\beta \mathrm{\beta ¬}$ is a $G$-acyclic object.

Then for every object $A\beta \mathrm{\pi }$ there is a spectral sequence $\left\{{E}_{p,q}^{r}\left(A\right){\right\}}_{r,p,q}$ called the Grothendieck spectral sequence whose ${E}_{2}$-page is the composite

${E}_{2}^{p,q}\left(A\right)={R}^{p}G\beta {R}^{q}F\left(A\right)$E^{p,q}_2(A) = R^p G \circ R^q F (A)

of the right derived functors of $F$ and $G$ in degrees $q$ and $p$, respectively and which is converging to to the derived functors ${R}^{n}\left(G\beta F\right)$ of the composite of $F$ and $G$:

${E}_{\beta }^{p,q}\left(A\right)\beta {G}^{p}{R}^{p+q}\left(G\beta F\right)\left(A\right)\phantom{\rule{thinmathspace}{0ex}}.$E^{p,q}_\infty(A) \simeq G^p R^{p+q}(G \circ F)(A) \,.

Moreover, this is natural in $A\beta \mathrm{\pi }$.

###### Proof

By assumption of enough injectives, we may find an injective resolution

$A\stackrel{{\beta }_{\mathrm{qi}}}{\beta }{C}^{\beta ’}$A \stackrel{\simeq_{qi}}{\to} C^\bullet

of $A$. Next, by the discussion at injective resolution β Existence and construction we may find a fully injective resolution of the chain complex $F\left({C}^{\beta ’}\right)$:

$0\beta F\left({C}^{\beta ’}\right)\beta {I}^{\beta ’,\beta ’}\phantom{\rule{thinmathspace}{0ex}},$0 \to F(C^\bullet) \to I^{\bullet, \bullet} \,,

where hence ${I}^{\beta ’,\beta ’}$ is a double complex of injective objects such that for each $n\beta \mathrm{\beta }$ the component $0\beta F\left({C}^{n}\right)\beta {I}^{n,\beta ’}$ is an ordinary injective resolution of $F\left({C}^{n}\right)\beta \mathrm{\beta ¬}$.

Thus we have the corresponding double complex $G\left({I}^{\beta ’,\beta ’}\right)$ in $\mathrm{\pi }$. The claim is that the Grothendieck spectral sequence is the spectral sequence of a double complex for $G\left({I}^{\beta ’,\beta ’}\right)$ equipped with the vertical-degree filtration $\left\{{}^{\mathrm{vert}}{E}_{p,q}^{r}\left(A\right)\right\}$:

${}^{\mathrm{vert}}{E}_{p,q}^{2}\left(A\right)\beta {R}^{p}G\left({R}^{q}F\left(A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.${}^{vert} E^2_{p,q}(A) \simeq R^p G (R^q F(A)) \,.

To see this, notice that by the assumption that ${I}^{\beta ’,\beta ’}$ is a fully injective projective resolution, the short exact sequences

$0\beta {B}^{q,p}\left(I\right)\beta {Z}^{q,p}\left(I\right)\beta {H}^{\left(}q,p\right)\left(I\right)\beta 0$0 \to B^{q,p}(I) \to Z^{q,p}(I) \to H^(q,p)(I) \to 0

are split (by the discussion there) and hence so is their image under any functor and hence in particular under $G$. Accordingly we have

$\begin{array}{rl}{}^{\mathrm{vert}}{E}_{1}^{p,q}& \beta {H}^{q}\left(G\left({I}^{\beta ’,p}\right)\right)\\ & \beta \left(G\left({Z}^{q,p}\right)\right)/\left(G\left({B}^{q,p}\right)\right)\\ & \beta G{H}^{q,p}\end{array}$\begin{aligned} {}^{vert}E^{p,q}_1 & \simeq H^q(G(I^{\bullet,p})) \\ & \simeq (G(Z^{q,p})) / (G(B^{q,p})) \\ & \simeq G H^{q,p} \end{aligned}

(the first two equivalences by general properties of the filtration spectral sequence, the last by the above splitness). Hence it follows that

$\begin{array}{rl}{}^{\mathrm{vert}}{E}_{2}^{p,q}& \beta {H}^{p}\left(G\left({H}^{q,\beta ’}\right)\right)\\ & \beta {R}^{p}G\left({R}^{q}F\left(A\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} {}^{vert}E^{p,q}_2 & \simeq H^p(G(H^{q,\bullet})) \\ & \simeq R^p G (R^q F (A)) \end{aligned} \,,

where in the last step we used that ${H}^{q,\beta ’}$ is be construction an injective resolution of ${H}^{q}\left(F\left({C}^{\beta ’}\right)\right)\beta {R}^{q}F\left(A\right)$ (using the $G$-acyclicity of $F\left({C}^{\beta ’}\right)$).

This establishes the spectral sequence and its second page as claimed. It remains to determine its convergence.

To that end, conside dually, the spectral sequence $\left\{{}^{\mathrm{hor}}{E}_{r}^{p,q}\right\}$ coming from the horizontal filtration on the double complex $G\left({I}^{\beta ’,\beta ’}\right)$. By the general properties of spectral sequence of a double complex this converges to the same value as the previous one. But for this latter spectral sequence we find

$\begin{array}{rl}{}^{\mathrm{hor}}{E}_{1}^{p,q}& \beta {H}^{q}\left(G{I}^{p,\beta ’}\right)\\ & \beta {R}^{q}G\left(F\left({C}^{p}\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} {}^{hor}E^{p,q}_1 & \simeq H^q(G I^{p,\bullet}) \\ & \simeq R^q G(F(C^p)) \end{aligned} \,,

the first equivalence by the general properties of filtration spectral sequences, the second then by the definition of right derived functors. But by assumption $F\left({C}^{p}\right)$ is $F$-acyclic and hence all these derived functors vanish in positive degree, so that

${}^{\mathrm{hor}}{E}_{1}^{p,q}\beta \left\{\begin{array}{cc}G\left(F\left({C}^{p}\right)\right)& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}q=0\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule{thinmathspace}{0ex}}.${}^{hor}E^{p,q}_1 \simeq \left\{ \array{ G(F(C^p)) & if\; q = 0 \\ 0 & otherwise } \right. \,.

Next, the ${E}_{2}$-page then contains just horizontal homology of $G\left(F\left({C}^{\beta ’}\right)\right)$ and this is by definition now the derived functor of the composite of $F$ with $G$:

${}^{\mathrm{hor}}{E}_{2}^{p,q}\beta \left\{\begin{array}{cc}{R}^{p}\left(G\beta F\right)& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}q=0\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule{thinmathspace}{0ex}}.${}^{hor}E^{p,q}_2 \simeq \left\{ \array{ R^p(G \circ F) & if \; q = 0 \\ 0 & otherwise } \right. \,.

Since this is concentrated in the $\left(q=0\right)$-row the spectral sequence of the horizontal filtration collapses here and hence

$\begin{array}{rl}{H}^{n}\left(\mathrm{Tot}\left(G\left({I}^{\beta ’,\beta ’}\right)\right)\right)& \beta {G}^{n}{H}^{n+0}\left(\mathrm{Tot}\left(G\left({I}^{\beta ’,\beta ’}\right)\right)\right)\\ & \beta {E}_{\beta }^{n,0}\end{array}$\begin{aligned} H^n(Tot(G(I^{\bullet,\bullet}))) &\simeq G^n H^{n+0}(Tot(G(I^{\bullet,\bullet}))) \\ & \simeq E^{n,0}_\infty \end{aligned}

So in conclusion we have

$\begin{array}{rl}{R}^{p}G\left({R}^{q}F\left(A\right)\right)& \beta {}^{\mathrm{vert}}{E}_{2}^{p,q}\\ & \beta {}^{\mathrm{vert}}{E}_{\beta }^{p,q}\\ & \beta {G}_{\mathrm{vert}}^{p}{H}^{p+q}\left(\mathrm{Tot}\left(G\left({I}^{\beta ’,\beta ’}\right)\right)\right)\\ & \beta {H}^{p+q}\left(\mathrm{Tot}\left(G\left({I}^{\beta ’,\beta ’}\right)\right)\right)\\ & \beta {G}_{\mathrm{hor}}^{p+q}{H}^{p+q}\left(\mathrm{Tot}\left(G\left({I}^{\beta ’,\beta ’}\right)\right)\right)\\ & \beta {}^{\mathrm{hor}}{E}_{\beta }^{p+q,0}\left(A\right)\\ & \beta {R}^{p+q}\left(G\beta F\right)\left(A\right)\end{array}$\begin{aligned} R^p G(R^q F(A)) & \simeq {}^{vert}E^{p,q}_2 \\ & \Rightarrow {}^{vert} E^{p,q}_\infty \\ & \simeq G^p_{vert} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq G^{p+q}_{hor} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq {}^{hor} E^{p+q,0}_\infty(A) \\ & \simeq R^{p+q}(G \circ F)(A) \end{aligned}

## Examples

Many other classes of spectral sequences are special cases of the Grothendieck spectral sequence, for instance the

## References

Leture notes include

• Jinhyun Park, Personal notes on Grothendieck spectral sequence (pdf)
Revised on October 29, 2012 23:25:22 by Urs Schreiber (131.174.188.167)