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

If the subobjects $Z_{\infty } = \bigcap Z_ r$ and $B_{\infty } = \bigcup B_ r$ of $E_1$ exist then we define the

*limit*^{1}of the spectral sequence to be the object $E_{\infty } = Z_{\infty }/B_{\infty }$.We say that the spectral sequence

*degenerates at $E_ r$*if the differentials $d_ r, d_{r + 1}, \ldots $ are all zero.

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