Lemma 12.23.7. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The associated spectral sequence

weakly converges to $H(K)$ if and only if for every $p \in \mathbf{Z}$ we have equality in equations (12.23.5.2) and (12.23.5.1),

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

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