Grothendieck spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors $$ G\circ F$$, from knowledge of the derived functors of $$F$$ and $$G$$. Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.

Statement
If $$F \colon\mathcal{A}\to\mathcal{B}$$ and $$G \colon \mathcal{B}\to\mathcal{C}$$ are two additive and left exact functors between abelian categories such that both $$\mathcal{A}$$ and $$\mathcal{B}$$ have enough injectives and $$F$$ takes injective objects to $$G$$-acyclic objects, then for each object $$A$$ of $$\mathcal{A}$$ there is a spectral sequence:


 * $$E_2^{pq} = ({\rm R}^p G \circ{\rm R}^q F)(A) \Longrightarrow {\rm R}^{p+q} (G\circ F)(A),$$

where $${\rm R}^p G$$ denotes the p-th right-derived functor of $$G$$, etc., and where the arrow '$$\Longrightarrow$$' means convergence of spectral sequences.

Five term exact sequence
The exact sequence of low degrees reads
 * $$0\to {\rm R}^1G(FA)\to {\rm R}^1(GF)(A) \to G({\rm R}^1F(A)) \to {\rm R}^2G(FA) \to {\rm R}^2(GF)(A).$$

The Leray spectral sequence
If $X$ and $Y$  are topological spaces, let $\mathcal{A} = \mathbf{Ab}(X)$  and $\mathcal{B} = \mathbf{Ab}(Y)$  be the category of sheaves of abelian groups on $X$  and $Y$, respectively.

For a continuous map $$f \colon X \to Y$$ there is the (left-exact) direct image functor $$f_* \colon \mathbf{Ab}(X) \to \mathbf{Ab}(Y)$$. We also have the global section functors


 * $$\Gamma_X \colon \mathbf{Ab}(X)\to \mathbf{Ab}$$ and $$\Gamma_Y \colon \mathbf{Ab}(Y) \to \mathbf {Ab}.$$

Then since $$\Gamma_Y \circ f_* = \Gamma_X$$ and the functors $$ f_*$$ and $$\Gamma_Y$$ satisfy the hypotheses (since the direct image functor has an exact left adjoint $$f^{-1}$$, pushforwards of injectives are injective and in particular 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 $$X$$.

Local-to-global Ext spectral sequence
There is a spectral sequence relating the global Ext and the sheaf Ext: let F, G be sheaves of modules over a ringed space $$(X, \mathcal{O})$$; e.g., a scheme. Then
 * $$E^{p,q}_2 = \operatorname{H}^p(X; \mathcal{E}xt^q_{\mathcal{O}}(F, G)) \Rightarrow \operatorname{Ext}^{p+q}_{\mathcal{O}}(F, G).$$

This is an instance of the Grothendieck spectral sequence: indeed,
 * $$R^p \Gamma(X, -) = \operatorname{H}^p(X, -)$$, $$R^q \mathcal{H}om_{\mathcal{O}}(F, -) = \mathcal{E}xt^q_{\mathcal{O}}(F, -)$$ and $$R^n \Gamma(X, \mathcal{H}om_{\mathcal{O}}(F, -)) = \operatorname{Ext}^n_{\mathcal{O}}(F, -)$$.

Moreover, $$\mathcal{H}om_{\mathcal{O}}(F, -)$$ sends injective $$\mathcal{O}$$-modules to flasque sheaves, which are $$\Gamma(X, -)$$-acyclic. Hence, the hypothesis is satisfied.

Derivation
We shall use the following lemma:

Proof: Let $$Z^n, B^{n+1}$$ be the kernel and the image of $$d: K^n \to K^{n+1}$$. We have
 * $$0 \to Z^n \to K^n \overset{d}\to B^{n+1} \to 0,$$

which splits. This implies each $$B^{n+1}$$ is injective. Next we look at
 * $$0 \to B^n \to Z^n \to H^n(K^{\bullet}) \to 0.$$

It splits, which implies the first part of the lemma, as well as the exactness of
 * $$0 \to G(B^n) \to G(Z^n) \to G(H^n(K^{\bullet})) \to 0.$$

Similarly we have (using the earlier splitting):
 * $$0 \to G(Z^n) \to G(K^n) \overset{G(d)} \to G(B^{n+1}) \to 0.$$

The second part now follows. $$\square$$

We now construct a spectral sequence. Let $$A^0 \to A^1 \to \cdots$$ be an injective resolution of A. Writing $$\phi^p$$ for $$F(A^p) \to F(A^{p+1})$$, we have:
 * $$0 \to \operatorname{ker} \phi^p \to F(A^p) \overset{\phi^p}\to \operatorname{im} \phi^p \to 0.$$

Take injective resolutions $$J^0 \to J^1 \to \cdots$$ and $$K^0 \to K^1 \to \cdots$$ of the first and the third nonzero terms. By the horseshoe lemma, their direct sum $$I^{p, \bullet} = J \oplus K$$ is an injective resolution of $$F(A^p)$$. Hence, we found an injective resolution of the complex:
 * $$0 \to F(A^{\bullet}) \to I^{\bullet, 0} \to I^{\bullet, 1} \to \cdots.$$

such that each row $$I^{0, q} \to I^{1, q} \to \cdots$$ satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

Now, the double complex $$E_0^{p, q} = G(I^{p, q})$$ gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,
 * $${}^{\prime \prime} E_1^{p, q} = H^q(G(I^{p, \bullet})) = R^q G(F(A^p))$$,

which is always zero unless q = 0 since $$F(A^p)$$ is G-acyclic by hypothesis. Hence, $${}^{\prime \prime} E_{2}^n = R^n (G \circ F) (A)$$ and $${}^{\prime \prime} E_2 = {}^{\prime \prime} E_{\infty}$$. On the other hand, by the definition and the lemma,
 * $${}^{\prime} E^{p, q}_1 = H^q(G(I^{\bullet, p})) = G(H^q(I^{\bullet, p})).$$

Since $$H^q(I^{\bullet, 0}) \to H^q(I^{\bullet, 1}) \to \cdots$$ is an injective resolution of $$H^q(F(A^{\bullet})) = R^q F(A)$$ (it is a resolution since its cohomology is trivial),
 * $${}^{\prime} E^{p, q}_2 = R^p G(R^qF(A)).$$

Since $${}^{\prime} E_r$$ and $${}^{\prime \prime} E_r$$ have the same limiting term, the proof is complete. $$\square$$

Computational Examples

 * Sharpe, Eric (2003). Lectures on D-branes and Sheaves (pages 18–19),