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

:$E_2^ = (^p G \circ^q F)(A) \Longrightarrow ^ (G\circ F)(A),$

where $^p G$ 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
:$0\to ^1G(FA)\to ^1(GF)(A) \to G(^1F(A)) \to ^2G(FA) \to ^2(GF)(A).$

Examples

The Leray spectral sequence

If $X$ and $Y$ are topological spaces, let $\mathcal = \mathbf(X)$ and $\mathcal = \mathbf(Y)$ 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 $f_* \colon \mathbf(X) \to \mathbf(Y)$.
We also have the global section functors

:$\Gamma_X \colon \mathbf(X)\to \mathbf$ and $\Gamma_Y \colon \mathbf(Y) \to \mathbf .$
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 $f^$, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:

:$H^p(Y,^q f_*\mathcal)\implies H^(X,\mathcal)$

for a sheaf $\mathcal$ 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 $(X, \mathcal)$; e.g., a scheme. Then
:$E^_2 = \operatorname^p(X; \mathcalxt^q_(F, G)) \Rightarrow \operatorname^_(F, G).$
This is an instance of the Grothendieck spectral sequence: indeed,
:$R^p \Gamma(X, -) = \operatorname^p(X, -)$, $R^q \mathcalom_(F, -) = \mathcalxt^q_(F, -)$ and $R^n \Gamma(X, \mathcalom_(F, -)) = \operatorname^n_(F, -)$.
Moreover, $\mathcalom_(F, -)$ sends injective $\mathcal$-modules to flasque sheaves, which are $\Gamma(X, -)$-acyclic. Hence, the hypothesis is satisfied.

Derivation

We shall use the following lemma:

Proof: Let $Z^n, B^$ be the kernel and the image of $d: K^n \to K^$. We have
:$0 \to Z^n \to K^n \overset\to B^ \to 0,$
which splits. This implies each $B^$ is injective. Next we look at
:$0 \to B^n \to Z^n \to H^n(K^) \to 0.$
It splits, which implies the first part of the lemma, as well as the exactness of
:$0 \to G(B^n) \to G(Z^n) \to G(H^n(K^)) \to 0.$
Similarly we have (using the earlier splitting):
:$0 \to G(Z^n) \to G(K^n) \overset \to G(B^) \to 0.$
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 $F(A^p) \to F(A^)$, we have:
:$0 \to \operatorname \phi^p \to F(A^p) \overset\to \operatorname \phi^p \to 0.$
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 $I^ = J \oplus K$ is an injective resolution of $F(A^p)$. Hence, we found an injective resolution of the complex:
:$0 \to F(A^) \to I^ \to I^ \to \cdots.$
such that each row $I^ \to I^ \to \cdots$ satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)

Now, the double complex $E_0^ = G(I^)$ gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,
:$^ E_1^ = H^q(G(I^)) = R^q G(F(A^p))$,
which is always zero unless ''q'' = 0 since $F(A^p)$ is ''G''-acyclic by hypothesis. Hence, $^ E_^n = R^n (G \circ F) (A)$ and $^ E_2 = ^ E_$. On the other hand, by the definition and the lemma,
:$^ E^_1 = H^q(G(I^)) = G(H^q(I^)).$
Since $H^q(I^) \to H^q(I^) \to \cdots$ is an injective resolution of $H^q(F(A^)) = R^q F(A)$ (it is a resolution since its cohomology is trivial),
:$^ E^_2 = R^p G(R^qF(A)).$
Since $^ E_r$ and $^ E_r$ have the same limiting term, the proof is complete. $\square$

Notes

References

*
*

Computational Examples

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

{{PlanetMath attribution, id=1095, title=Grothendieck spectral sequence

Spectral sequences