The Stacks project

Definition 12.20.2. Let $\mathcal{A}$ be an abelian category. Let $(E_ r, d_ r)_{r \geq 1}$ be a spectral sequence.

  1. If the subobjects $Z_{\infty } = \bigcap Z_ r$ and $B_{\infty } = \bigcup B_ r$ of $E_1$ exist then we define the limit1 of the spectral sequence to be the object $E_{\infty } = Z_{\infty }/B_{\infty }$.

  2. We say that the spectral sequence degenerates at $E_ r$ if the differentials $d_ r, d_{r + 1}, \ldots $ are all zero.

[1] This notation is not universally accepted. In some references an additional pair of subobjects $Z_\infty $ and $B_\infty $ of $E_1$ such that $0 = B_1 \subset B_2 \subset \ldots \subset B_\infty \subset Z_\infty \subset \ldots \subset Z_2 \subset Z_1 = E_1$ is part of the data comprising a spectral sequence!

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