Differentials on the second page of the spectral sequence of a first quadrant double complex

Question

Suppose we have some (homological) double complex $\{E_{pq}\}$ with $p$ labelling the row and $q$ the column (is this standard or not?). Taking the homology of the vertical maps, it's easy enough to obtain the next page of the spectral sequence, with horizontal arrows induced by the morphisms of chain complexes (the columns in the zeroth page). My confusion is on how to obtain the differential on the second page. This should move two to the left and one up. That is, we want morphisms $$ d_{pq}: H_{q}\big(H_{p}(E_{\bullet q}) \big) \longrightarrow H_{q-2} \big(H_{p+1}( E_{\bullet q-2} )\big). $$ I'm having a lot of trouble seeing how this arises. Most references either leave it as an exercise, or claim it is obvious. It seems to be begging to use the Snake lemma, but I haven't been able to figure out what short exact sequences to take as rows for the snake diagram. Can anyone point me in the right direction?

Answer 1

I will answer this for cohomology. Let the 0-th page be a double complex $E_{0}^{p,q}.$ The first page will be the cohomology of the vertical differentials. Given a class $\alpha\in E^{p,q}_{2}$, it can be represented by a cocycle $[x]\in E_{1}^{p,q}$ where $x$ is in $E_{0}^{p,q}$, such that $$\begin{array}{ll} (1). & d_{\uparrow}(x)=0\\ (2).& d_{\rightarrow}(x)=d_{\uparrow}(y)\text{ for some }y\in E_{0}^{p+1,q-1} \end{array}.$$ Now $d_{\rightarrow}(y)\in E_{0}^{p+2,q-1}$. By (2), $d_{\uparrow}(d_{\rightarrow}(y))=\pm d_{\rightarrow}(d_{\rightarrow}(x))=0$ ($\pm$ depends on your convention of the commutativity of your double complex). Hence $d_{\rightarrow}(y)$ induces a cocycle $[d_{\rightarrow}(y)]\in E_{1}^{p+2,q-1}$. Since $d_{\rightarrow}(d_{\rightarrow}(y))=0$, $[d_{\rightarrow}(y)]$ is a cocycle in $E_{1}^{p+2,q-1}$ and therefore induces a class $\beta\in E_{2}^{p,q}.$ You define $E_{2}^{p,q}\rightarrow E_{2}^{p+2,q-1}$ by $$\alpha\mapsto \pm\beta$$ depending on your convention of the commutativity of your double complex. The rest of the detail can be found at A User's Guide to Spectral Sequences by John McCleary, theorem 2.6 and section 2.4.