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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 $𝒜,ℬ,𝒞$ be abelian categories and let $F:𝒜\to ℬ$ and $G:ℬ\to 𝒞$ be left exact additive functors. Assume that $𝒜,ℬ$ 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 $RF:D^{+}(A)\to D^{+}(B)$, $RG:D^{+}(B)\to D^{+}(C)$ and $R(G\circ F):D^{+}(A)\to D^{+}(C)$ are defined and the natural morphism $R(G\circ F)\to RG\circ RF$ is an isomorphism.

Theorem (Tohoku)

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

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

$$E_2^{p,q}(A)=R^{p}G\circ R^{q}F(A)$$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 $G$:

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

Moreover, this is natural in $A\in 𝒜$.

Proof

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

$$A\stackrel{{\simeq }_{\mathrm{qi}}}{\to }{C}^{•}$$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 $F(C^{•})$:

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

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

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

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

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

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

$$\begin{array}{rl}{}^{\mathrm{vert}}{E}_1^{p,q}& \simeq H^q(G(I^{•,p}))\\ & \simeq (G(Z^{q,p}))/(G(B^{q,p}))\\ & \simeq G{H}^{q,p}\end{array}$$\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

$$\begin{array}{rl}{}^{\mathrm{vert}}{E}_2^{p,q}& \simeq H^p(G(H^{q,•}))\\ & \simeq R^{p}G(R^{q}F(A))\end{array},$$\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 $H^{q,•}$ is be construction an injective resolution of $H^q(F(C^{•}))\simeq R^{q}F(A)$ (using the $G$-acyclicity of $F(C^{•})$).

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 $\{{}^{\mathrm{hor}}{E}_r^{p,q}\}$ coming from the horizontal filtration on the double complex $G(I^{•,•})$. 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

$$\begin{array}{rl}{}^{\mathrm{hor}}{E}_1^{p,q}& \simeq H^q(G{I}^{p,•})\\ & \simeq R^{q}G(F(C^p))\end{array},$$\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 $F$-acyclic and hence all these derived functors vanish in positive degree, so that

$${}^{\mathrm{hor}}{E}_1^{p,q}\simeq \{\begin{array}{cc}G(F(C^p))& \mathrm{if}q=0\\ 0& \mathrm{otherwise}\end{array}.$${}^{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 $G(F(C^{•}))$ and this is by definition now the derived functor of the composite of $F$ with $G$:

$${}^{\mathrm{hor}}{E}_2^{p,q}\simeq \{\begin{array}{cc}{R}^p(G\circ F)& \mathrm{if}q=0\\ 0& \mathrm{otherwise}\end{array}.$${}^{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

$$\begin{array}{rl}{H}^n(\mathrm{Tot}(G(I^{•,•})))& \simeq G^n{H}^{n+0}(\mathrm{Tot}(G(I^{•,•})))\\ & \simeq E_{\mathrm{\infty }}^{n,0}\end{array}$$\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

$$\begin{array}{rl}{R}^{p}G(R^{q}F(A))& \simeq {}^{\mathrm{vert}}{E}_2^{p,q}\\ & \Rightarrow {}^{\mathrm{vert}}{E}_{\mathrm{\infty }}^{p,q}\\ & \simeq G_{\mathrm{vert}}^p{H}^{p+q}(\mathrm{Tot}(G(I^{•,•})))\\ & \simeq H^{p+q}(\mathrm{Tot}(G(I^{•,•})))\\ & \simeq G_{\mathrm{hor}}^{p+q}{H}^{p+q}(\mathrm{Tot}(G(I^{•,•})))\\ & \simeq {}^{\mathrm{hor}}{E}_{\mathrm{\infty }}^{p+q,0}(A)\\ & \simeq R^{p+q}(G\circ F)(A)\end{array}$$\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

• Hochschild-Serre spectral sequence

• Leray spectral sequence

• base change spectral sequence

References

Leture notes include

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