The Stacks project

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. $\square$

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