entering differentials if for every fixed bidegree \((p, q)\text{,}\) all but finitely many of the differentials \(d^r\) landing in \(E^r_{p,q}\) vanish;
exiting differentials if for every fixed bidegree \((p, q)\text{,}\) all but finitely many of the differentials \(d^r\) originating from \(E^r_{p,q}\) vanish;
In this course we will essentially only encounter finite spectral sequences. The crucial consequence of finiteness is that for any fixed bidegree \((p, q)\) there exists \(R = R(p, q)\) such that
\begin{equation*}
E^{r+1}_{p,q} \;\cong\; E^r_{p,q} \qquad \text{for all } r \geq R,
\end{equation*}
because both incoming and outgoing differentials at \((p, q)\) vanish for \(r \geq R\text{.}\) The common stable value is the limit page, denoted
We say that a spectral sequence collapses on page \(R\) if \(d^r = 0\) for all \(r \geq R\text{.}\) In this case \(E^\infty_{p,q} = E^R_{p,q}\) uniformly in \((p, q)\text{.}\)
The limit page can in fact be defined for any spectral sequence, not necessarily finite. Writing \(Z E^r_{p,q} := \ker(d^r_{p,q})\text{,}\) an element of \(E^\infty_{p,q}\) is represented by a sequence \(a_{r_0} \in E^{r_0}_{p,q}\) that is a cycle, \(a_{r_0+1} = [a_{r_0}] \in E^{r_0+1}_{p,q}\) is again a cycle, and so on; two such sequences agree if their classes do on some sufficiently late page. For finite spectral sequences this recovers the definition above.
