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Tag: 012U

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Definition 11.18.7. 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 $(K^\bullet, F)$ converges if $\text{gr} H^n(K^\bullet) = \bigoplus_{p + q = n} E_{\infty}^{p, q}$ for every $n \in \mathbf{Z}$.

\begin{definition}
\label{definition-filtered-complex-ss-converges}
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 $(K^\bullet, F)$
{\it converges} if
$\text{gr} H^n(K^\bullet) = \bigoplus_{p + q = n} E_{\infty}^{p, q}$
for every $n \in \mathbf{Z}$.
\end{definition}


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