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The Stacks project

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(\text{Tot}(K^{\bullet , \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^*(\text{Tot}(K^{\bullet , \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(\text{Tot}(K^{\bullet , \bullet })) is in \mathcal{C}. Similarly for ({}''E_ r, {}''d_ r)_{r \geq 0}.

Proof. Follows immediately from Lemma 12.24.11. \square


Comments (2)

Comment #7058 by Xiaolong Liu on

We should replace "" in (2) by "".

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