Definition 12.24.9. Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet , F)$ be a filtered complex of $\mathcal{A}$. We say the spectral sequence associated to $(K^\bullet , F)$

*weakly converges to $H^*(K^\bullet )$*if $\text{gr}^ pH^ n(K^\bullet ) = E_{\infty }^{p, n - p}$ via Lemma 12.24.6 for all $p, n \in \mathbf{Z}$,*abuts to $H^*(K^\bullet )$*if it weakly converges to $H^*(K^\bullet )$ and $\bigcap _ p F^ pH^ n(K^\bullet ) = 0$ and $\bigcup _ p F^ p H^ n(K^\bullet ) = H^ n(K^\bullet )$ for all $n$,*converges to $H^*(K^\bullet )$*if it is regular, abuts to $H^*(K^\bullet )$, and $H^ n(K^\bullet ) = \mathop{\mathrm{lim}}\nolimits _ p H^ n(K^\bullet )/F^ pH^ n(K^\bullet )$.

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