Section 7.2 Convergence
But back to spectral sequences. For now, let us stick to cohomological spectral sequences; any general statement we make about these can easily be translate into a statement about homological spectral sequences.
Definition 7.31.
Definition 8.30. We say that a spectral sequence \(E=\left(E^{r}\right)\) is bounded if for every \(n\) there are only finitely many nonzero terms of total degree \(n\text{,}\) meaning that there are only finitely many pairs \((p, q)\) with \(n=p+q\) such that \(E_{p, q}^{r} \neq 0\) for all \(r\text{.}\)
Most spectral sequences one deals with end up being bounded, so we will focus only on the case of bounded spectral sequences. This guarantees that what we are about to do makes sense.
From the data in a spectral sequence \(E\) we define a limiting page \(E_{\infty}\text{,}\) which can often be identified with some interesting object (for example, the homology of a complex we care about). These \(E_{\infty}\) pages are sort of the whole point of the spectral sequence business, as they contain (pieces of) the information we want to compute.
Construction 8.32 ( \(E_{\infty}\) page). Set
\begin{equation*}
B_{0}:=0 \quad \text { and } \quad Z_{0}=E_{0}
\end{equation*}
so that \(E_{0}=Z_{0} / B_{0}\text{.}\) At each stage \(r\text{,}\) given \(B_{r} \subseteq Z_{r}\) such that \(E_{r} \cong Z_{r} / B_{r}\text{,}\) we define
\begin{equation*}
Z_{r+1}:=\operatorname{ker}\left(Z_{r} \longrightarrow Z_{r} / B_{r}=E_{r} \stackrel{d_{r}}{\longrightarrow} E_{r}=Z_{r} / B_{r}\right)
\end{equation*}
and \(B_{r+1}\) such that \(B_{r} \subseteq B_{r+1} \subseteq Z_{r}\) and
\begin{equation*}
B_{r+1} / B_{r}:=\operatorname{im}\left(Z_{r} \longrightarrow Z_{r} / B_{r}=E_{r} \stackrel{d_{r}}{\longrightarrow} E_{r}=Z_{r} / B_{r}\right)
\end{equation*}
At each stage,
\begin{equation*}
B_{r} \subseteq B_{r+1} \subseteq Z_{r+1} \subseteq Z_{r}
\end{equation*}
and
\begin{equation*}
E_{r+1}=H\left(E_{r}\right)=\cong Z_{r+1} / B_{r+1}
\end{equation*}
Thus we get chains
\begin{equation*}
0=B_{0} \subseteq B_{1} \subseteq B_{2} \subseteq \cdots \subseteq Z_{r} \subseteq Z_{r-1} \subseteq \cdots \subseteq Z_{1} \subseteq Z_{0}=E_{0}
\end{equation*}
We say that \(Z_{r}\) consists of the elements that survive until stage \(r\text{,}\) while \(B_{r}\) consists of the elements that are in the image of the differentials by stage \(r\text{.}\) We define
\begin{equation*}
B_{\infty}:=\bigcup_{i} B_{i} \quad \text { and } \quad B_{\infty}:=\bigcap_{i} Z_{i}
\end{equation*}
The elements in \(B_{\infty}^{p, q}\) are the (classes of) those elements in \(E_{0}^{p, q}\) that are in the image of the differential at some stage; we say these are the elements that are eventually bound. The elements in \(Z_{\infty}^{p, q}\) are the (classes of) those elements in \(E_{0}^{p, q}\) that are in the kernel of all the differentials at all stages, so they survive forever or live forever.
Definition 7.33.
Definition 8.33. Given a homological spectral sequence \(E=\left(E_{r}\right)\text{,}\) the \(E_{\infty}\) page is the bigraded module given by
\begin{equation*}
E_{\infty}^{p, q}:=Z_{\infty}^{p, q} / B_{\infty}^{p, q}
\end{equation*}
Similarly, given a cohomological spectral sequence \(\left(E^{r}\right)\) one can define \(Z^{r}\) and \(B^{r}, Z^{\infty}\) and \(B^{\infty}\text{,}\) and the \(E^{\infty}\) page
\begin{equation*}
E_{p, q}^{\infty}:=Z_{p, q}^{\infty} / B_{p, q}^{\infty}
\end{equation*}
Lemma 7.34.
Lemma 8.34. Let \(E=\left(E_{r}\right)\) be a spectral sequence. We have
\begin{equation*}
E_{r+1}=E_{r} \Longleftrightarrow Z_{r+1}=Z_{r} \text { and } B_{r+1}=B_{r} \text {. }
\end{equation*}
Proof.
Proof. In general, if \(X / Y\) is a subquotient of \(Z\text{,}\) we have \(Y \subseteq X \subseteq Z\text{,}\) so \(X / Y=Z\) if and only if \(Y=0\) and \(X=Z\text{.}\) If \(E_{r+1}=E_{r}\text{,}\) then
\begin{equation*}
Z_{r+1} / B_{r+1}=E_{r+1}=E_{r}=Z_{r} / B_{r},
\end{equation*}
so \(B_{r+1}=0\) in \(Z_{r} / B_{r}\text{,}\) so we must have \(B_{r+1}=B_{r}\text{.}\) But then
\begin{equation*}
Z_{r+1} / B_{r}=Z_{r} / B_{r}
\end{equation*}
so \(Z_{r+1}=Z_{r}\text{.}\)
Conversely, if \(Z_{r+1}=Z_{r}\) and \(B_{r+1}=B_{r}\text{,}\) then
\begin{equation*}
E_{r+1}=Z_{r+1} / B_{r+1}=Z_{r} / B_{r}=E_{r} .
\end{equation*}
This \(E^{\infty}\) is easier to compute in the following special cases.
Definition 7.35.
Definition 8.35. We say that a spectral sequence \(E=\left(E_{r}\right)\) degenerates at the \(n\)th page if \(d_{r}=0\) for all \(r \geqslant n\text{.}\)
Example 7.37.
Example 8.37. Let \(E\) be a spectral sequence. If the \(r\) th page is concentrated in one row or one columnn, we say that the spectral sequence collapses at the \(r\) th page. Notice that in such situations the spectral sequence will automatically degenerate at the \(r\) th page.
Definition 7.38.
Definition 8.38. Le \(H\) be a graded \(R\)-module. We say that a spectral sequence \(E=\left(E^{r}\right)\) converges to the graded \(R\)-module \(H=\left(H_{n}\right)\text{,}\) and denote it by
\begin{equation*}
E_{2}^{p, q} \Longrightarrow H_{p+q}
\end{equation*}
if there exists a bounded filtration \(F^{\bullet}\) for \(H\) such that
\begin{equation*}
E_{\infty}^{p, q} \cong F^{p} H_{p+q} / F^{p-1} H_{p+q} \quad \text { for all } p, q
\end{equation*}
Alternatively, we may write \(E_{r}^{p, q} \Longrightarrow H_{p+q}\) for some other fixed choice of \(r\) besides \(r=2\text{.}\) We have the basic definitions but we haven’t yet seen any examples. This is by design; to give a good example of a spectral sequence we need to do a bit more work than to give a starting example for a run of the mill definition. In the next few sections we will discuss some of the ways in which spectral sequences arise. There are, however, many interesting spectral sequences one would discuss in a first course that we do not have the time to cover. We strongly encourage the reader to seek out better sources, such as those we listed in the beginning of the chapter.
