Lemma 12.24.10. Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet , F)$ be a filtered complex of $\mathcal{A}$. The associated spectral sequence

1. weakly converges to $H^*(K^\bullet )$ if and only if for every $p, q \in \mathbf{Z}$ we have equality in equations (12.24.6.2) and (12.24.6.1),

2. abuts to $H^*(K)$ if and only if it weakly converges to $H^*(K^\bullet )$ and we have $\bigcap _ p (\mathop{\mathrm{Ker}}(d) \cap F^ pK^ n + \mathop{\mathrm{Im}}(d) \cap K^ n) = \mathop{\mathrm{Im}}(d) \cap K^ n$ and $\bigcup _ p (\mathop{\mathrm{Ker}}(d) \cap F^ pK^ n + \mathop{\mathrm{Im}}(d) \cap K^ n) = \mathop{\mathrm{Ker}}(d) \cap K^ n$.

Proof. Immediate from the discussions above. $\square$

Comment #1279 by on

I was trying to cook up a slogan for this tag, but it is pretty hard, because of the equations. Is a slogan like "Conditions for the convergence of the spectral sequence of a filtered complex" suitable in cases like this? It is pretty vague, but describing the equations in words is not doable either.

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