nLab Grothendieck spectral sequence

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 $\mathcal{A},\mathcal{B},\mathcal{C}$ be abelian categories and let $F \colon \mathcal{A}\to \mathcal{B}$ and $G \colon \mathcal{B}\to \mathcal{C}$ be left exact additive functors. Assume that $\mathcal{A}, \mathcal{B}$ have enough injectives.

Theorem (Tohoku)

Write $R_F\subset \mathrm{Ob} A$ and $R_G\subset\mathrm{Ob} B$ for the classes of objects adapted to $F$ and $G$ respectively, and let furthermore $F(R_A)\subset R_B$. Then the derived functors $R F:D^+(A)\to D^+(B)$, $R G:D^+(B)\to D^+(C)$ and $R(G\circ F):D^+(A)\to D^+(C)$ are defined and the natural morphism $R(G\circ F)\to R G\circ R F$ is an isomorphism.

Theorem (Tohoku)

In the above situation, assume that for every injective object $I \in \mathcal{A}$ the object $F(I) \in \mathcal{B}$ is a $G$-acyclic object.

Then for every object $A \in \mathcal{A}$ there is a spectral sequence $\{E^r_{p,q}(A)\}_{r,p,q}$ called the Grothendieck spectral sequence whose $E_2$-page is the composite

$$E^{p,q}_2(A) = R^p G \circ R^q F (A)$$

of the right derived functors of $F$ and $G$ in degrees $q$ and $p$, respectively and which is converging to to the derived functors $R^n(G\circ F)$ of the composite of $F$ and $G$:

$$E^{p,q}_\infty(A) \simeq R^{p+q}(G \circ F)(A) \,.$$

Moreover, this is natural in $A \in \mathcal{A}$.

Proof

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

$$A \stackrel{\simeq_{qi}}{\to} C^\bullet$$

of $A$. Next, by the discussion at injective resolution – Existence and construction we may find a fully injective resolution of the chain complex $F(C^\bullet)$:

$$0 \to F(C^\bullet) \to I^{\bullet, \bullet} \,,$$

where hence $I^{\bullet, \bullet}$ is a double complex of injective objects such that for each $n \in \mathbb{N}$ the component $0 \to F(C^n) \to I^{n,\bullet}$ is an ordinary injective resolution of $F(C^n) \in \mathcal{B}$.

Thus we have the corresponding double complex $G(I^{\bullet,\bullet})$ in $\mathcal{C}$. The claim is that the Grothendieck spectral sequence is the spectral sequence of a double complex for $G(I^{\bullet, \bullet})$ equipped with the vertical-degree filtration $\{{}^{vert}E^r_{p,q}(A)\}$:

$${}^{vert} E^2_{p,q}(A) \simeq R^p G (R^q F(A)) \,.$$

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

$$0 \to B^{q,p}(I) \to Z^{q,p}(I) \to H^{q,p}(I) \to 0$$

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

$$\begin{aligned} {}^{vert}E^{p,q}_1 & \simeq H^q(G(I^{\bullet,p})) \\ & \simeq (G(Z^{q,p})) / (G(B^{q,p})) \\ & \simeq G H^{q,p} \end{aligned}$$

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

$$\begin{aligned} {}^{vert}E^{p,q}_2 & \simeq H^p(G(H^{q,\bullet})) \\ & \simeq R^p G (R^q F (A)) \end{aligned} \,,$$

where in the last step we used that $H^{q,\bullet}$ is be construction an injective resolution of $H^q(F(C^\bullet)) \simeq R^q F(A)$ (using the $G$-acyclicity of $F(C^\bullet)$).

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

To that end, consider dually, the spectral sequence $\{{}^{hor}E^{p,q}_r\}$ coming from the horizontal filtration on the double complex $G(I^{\bullet, \bullet})$. 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

$$\begin{aligned} {}^{hor}E^{p,q}_1 & \simeq H^q(G I^{p,\bullet}) \\ & \simeq R^q G(F(C^p)) \end{aligned} \,,$$

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

$${}^{hor}E^{p,q}_1 \simeq \left\{ \array{ G(F(C^p)) & if\; q = 0 \\ 0 & otherwise } \right. \,.$$

Next, the $E_2$-page then contains just horizontal homology of $G(F(C^\bullet))$ and this is by definition now the derived functor of the composite of $F$ with $G$:

$${}^{hor}E^{p,q}_2 \simeq \left\{ \array{ R^p(G \circ F) & if \; q = 0 \\ 0 & otherwise } \right. \,.$$

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

$$\begin{aligned} H^n(Tot(G(I^{\bullet,\bullet}))) &\simeq G^n H^{n+0}(Tot(G(I^{\bullet,\bullet}))) \\ & \simeq E^{n,0}_\infty \end{aligned}$$

So in conclusion we have

$$\begin{aligned} R^p G(R^q F(A)) & \simeq {}^{vert}E^{p,q}_2 \\ & \Rightarrow {}^{vert} E^{p,q}_\infty \\ & \simeq G^p_{vert} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq G^{p+q}_{hor} H^{p+q}(Tot(G(I^{\bullet, \bullet}))) \\ & \simeq {}^{hor} E^{p+q,0}_\infty(A) \\ & \simeq R^{p+q}(G \circ F)(A) \end{aligned}$$

Examples

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

• Hochschild-Serre spectral sequence

• Leray spectral sequence

• base change spectral sequence

References

Leture notes include

• James Milne, section 10 of Lectures on Étale Cohomology

• Jinhyun Park, Personal notes on Grothendieck spectral sequence (pdf)