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Definition 12.23.6. Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. We say the spectral sequence associated to $(K, F, d)$

  1. weakly converges to $H(K)$ if $\text{gr}H(K) = E_{\infty }$ via Lemma 12.23.5,

  2. abuts to $H(K)$ if it weakly converges to $H(K)$ and we have $\bigcap F^ pH(K) = 0$ and $\bigcup F^ pH(K) = H(K)$,

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