Lemma 12.24.8. In the situation of Definition 12.24.7. Let $Z_ r^{p, q}, B_ r^{p, q} \subset E_{r_0}^{p, q}$ be the $(p, q)$-graded parts of $Z_ r, B_ r$ defined as in Section 12.20.

1. The spectral sequence is regular if and only if for all $p, q$ there exists an $r = r(p, q)$ such that $Z_ r^{p, q} = Z_{r + 1}^{p, q} = \ldots$

2. The spectral sequence is coregular if and only if for all $p, q$ there exists an $r = r(p, q)$ such that $B_ r^{p, q} = B_{r + 1}^{p, q} = \ldots$

3. The spectral sequence is bounded if and only if it is both bounded below and bounded above.

4. If the spectral sequence is bounded below, then it is regular.

5. If the spectral sequence is bounded above, then it is coregular.

Proof. Omitted. Hint: If $E_ r^{p, q} = 0$, then we have $E_{r'}^{p, q} = 0$ for all $r' \geq r$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).