Lemma 12.25.3. Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet , \bullet }$ be a double complex. Assume that for every $n \in \mathbf{Z}$ there are only finitely many nonzero $K^{p, q}$ with $p + q = n$. Then

1. the two spectral sequences associated to $K^{\bullet , \bullet }$ are bounded,

2. the filtrations $F_ I$, $F_{II}$ on each $H^ n(K^\bullet )$ are finite,

3. the spectral sequences $({}'E_ r, {}'d_ r)_{r \geq 0}$ and $({}''E_ r, {}''d_ r)_{r \geq 0}$ converge to $H^*(sK^\bullet )$,

4. if $\mathcal{C} \subset \mathcal{A}$ is a weak Serre subcategory and for some $r$ we have ${}'E_ r^{p, q} \in \mathcal{C}$ for all $p, q \in \mathbf{Z}$, then $H^ n(sK^\bullet )$ is in $\mathcal{C}$. Similarly for $({}''E_ r, {}''d_ r)_{r \geq 0}$.

Proof. Follows immediately from Lemma 12.24.11. $\square$

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