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Section 18.1 What is a Spectral Sequence?

Spectral sequences come in various flavours (homological vs. cohomological) and with various gradings (homological/cohomological Serre grading, homological/cohomological Adams grading). We will work with the homological Serre grading throughout this chapter; other conventions are obtained by reindexing. Roughly, a spectral sequence (of abelian groups) consists of three pieces of data.
  1. Bigraded abelian groups. For each integer \(r \geq 1\) and each pair \((p, q) \in \mathbb{Z} \times \mathbb{Z}\text{,}\) an abelian group \(E^r_{p,q}\text{.}\) For a fixed \(r\) we picture the collection \(\{E^r_{p,q}\}_{p,q}\) as sitting in a planar grid:
    This is called the \(r\)th page of the spectral sequence. It is common to think of a spectral sequence as a book whose \(r\)th page is the picture above.
  2. Differentials on each page. Maps
    \begin{equation*} d^r \colon E^r_{*,*} \;\longrightarrow\; E^r_{*,*}, \qquad d^r \circ d^r = 0. \end{equation*}
    The bidegree of \(d^r\) depends on the grading convention. For the homological Serre grading,
    \begin{equation*} d^r \colon E^r_{p,q} \;\longrightarrow\; E^r_{p-r,\, q+r-1}. \end{equation*}
    Pictorially \(d^r\) moves \(r\) to the left and \(r-1\) up.
  3. Page-to-page isomorphisms. For each \(r \geq 1\text{,}\)
    \begin{equation*} E^{r+1}_{p,q} \;\cong\; H\bigl(E^r_{p,q},\, d^r\bigr) \;=\; \frac{\ker\bigl(d^r \colon E^r_{p,q} \to E^r_{p-r,\,q+r-1}\bigr)}{\mathrm{im}\bigl(d^r \colon E^r_{p+r,\,q-r+1} \to E^r_{p,q}\bigr)}. \end{equation*}
    That is, we pass from the \(r\)th page to the \((r+1)\)th page by taking homology with respect to the differential.
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