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?

  • $\begingroup$ Ravi Vakil's online notes on algebraic geometry explain this construction pretty well, if I'm remembering right... $\endgroup$ – Lorenzo Aug 6 '18 at 8:06
  • $\begingroup$ @Lorenzo Unfortunately, you are incorrect. Ravi leaves it as an exercise $\endgroup$ – Harry Gindi Aug 17 '18 at 4:59

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