20.29 Cohomology of filtered complexes
Filtered complexes of sheaves frequently come up in a natural fashion when studying cohomology of algebraic varieties, for example the de Rham complex comes with its Hodge filtration. In this sectionwe use the very general Injectives, Lemma 19.13.7 to find construct spectral sequences on cohomology and we relate these to previously constructed spectral sequences.
Lemma 20.29.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a filtered complex of $\mathcal{O}_ X$-modules. There exists a canonical spectral sequence $(E_ r, \text{d}_ r)_{r \geq 1}$ of bigraded $\Gamma (X, \mathcal{O}_ X)$-modules with $d_ r$ of bidegree $(r, -r + 1)$ and
\[ E_1^{p, q} = H^{p + q}(X, \text{gr}^ p\mathcal{F}^\bullet ) \]
If for every $n$ we have
\[ H^ n(X, F^ p\mathcal{F}^\bullet ) = 0\text{ for }p \gg 0 \quad \text{and}\quad H^ n(X, F^ p\mathcal{F}^\bullet ) = H^ n(X, \mathcal{F}^\bullet )\text{ for }p \ll 0 \]
then the spectral sequence is bounded and converges to $H^*(X, \mathcal{F}^\bullet )$.
Proof.
(For a proof in case the complex is a bounded below complex of modules with finite filtrations, see the remark below.) Choose an map of filtered complexes $j : \mathcal{F}^\bullet \to \mathcal{J}^\bullet $ as in Injectives, Lemma 19.13.7. The spectral sequence is the spectral sequence of Homology, Section 12.24 associated to the filtered complex
\[ \Gamma (X, \mathcal{J}^\bullet ) \quad \text{with}\quad F^ p\Gamma (X, \mathcal{J}^\bullet ) = \Gamma (X, F^ p\mathcal{J}^\bullet ) \]
Since cohomology is computed by evaluating on K-injective representatives we see that the $E_1$ page is as stated in the lemma. The convergence and boundedness under the stated conditions follows from Homology, Lemma 12.24.13.
$\square$
Example 20.29.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. We can apply Lemma 20.29.1 with $F^ p\mathcal{F}^\bullet = \tau _{\leq -p}\mathcal{F}^\bullet $. (If $\mathcal{F}^\bullet $ is bounded below we can use Remark 20.29.2.) Then we get a spectral sequence
\[ E_1^{p, q} = H^{p + q}(X, H^{-p}(\mathcal{F}^\bullet )[p]) = H^{2p + q}(X, H^{-p}(\mathcal{F}^\bullet )) \]
After renumbering $p = -j$ and $q = i + 2j$ we find that for any $K \in D(\mathcal{O}_ X)$ there is a spectral sequence $(E'_ r, d'_ r)_{r \geq 2}$ of bigraded modules with $d'_ r$ of bidegree $(r, -r + 1)$, with
\[ (E'_2)^{i, j} = H^ i(X, H^ j(K)) \]
If $K$ is bounded below (for example), then this spectral sequence is bounded and converges to $H^{i + j}(X, K)$. In the bounded below case this spectral sequence is an example of the second spectral sequence of Derived Categories, Lemma 13.21.3 (constructed using Cartan-Eilenberg resolutions).
Example 20.29.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. We can apply Lemma 20.29.1 with $F^ p\mathcal{F}^\bullet = \sigma _{\geq p}\mathcal{F}^\bullet $. Then we get a spectral sequence
\[ E_1^{p, q} = H^{p + q}(X, \mathcal{F}^ p[-p]) = H^ q(X, \mathcal{F}^ p) \]
If $\mathcal{F}^\bullet $ is bounded below, then
we can use Remark 20.29.2 to construct this spectral sequence,
the spectral sequence is bounded and converges to $H^{i + j}(X, \mathcal{F}^\bullet )$, and
the spectral sequence is equal to the first spectral sequence of Derived Categories, Lemma 13.21.3 (constructed using Cartan-Eilenberg resolutions).
Lemma 20.29.5. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet $ be a filtered complex of $\mathcal{O}_ X$-modules. There exists a canonical spectral sequence $(E_ r, \text{d}_ r)_{r \geq 1}$ of bigraded $\mathcal{O}_ Y$-modules with $d_ r$ of bidegree $(r, -r + 1)$ and
\[ E_1^{p, q} = R^{p + q}f_*\text{gr}^ p\mathcal{F}^\bullet \]
If for every $n$ we have
\[ R^ nf_*F^ p\mathcal{F}^\bullet = 0 \text{ for }p \gg 0 \quad \text{and}\quad R^ nf_*F^ p\mathcal{F}^\bullet = R^ nf_*\mathcal{F}^\bullet \text{ for }p \ll 0 \]
then the spectral sequence is bounded and converges to $Rf_*\mathcal{F}^\bullet $.
Proof.
The proof is exactly the same as the proof of Lemma 20.29.1.
$\square$
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