structural:expansion, structural:summarization, information:classification, structural:reformatting
Spectral sequence

Fix an abelian category, such as a category of modules over a ring. A spectral sequence is a choice of a nonnegative integer r and a collection of three sequences:

A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer. We have three indices, r, p, and q. For each r, imagine that we have a sheet of graph paper. On this sheet, we will take p to be the horizontal direction and q to be the vertical direction. At each lattice point we have the object E r p, q {\displaystyle E_{r}^{p,q}}.

It is very common for n = p + q to be another natural index in the spectral sequence. n runs diagonally, northwest to southeast, across each sheet. In the homological case, the differentials have bidegree (− r, r − 1), so they decrease n by one. In the cohomological case, n is increased by one. When r is zero, the differential moves objects one space down or up. This is similar to the differential on a chain complex. When r is one, the differential moves objects one space to the left or right. When r is two, the differential moves objects just like a knight 's move in chess. For higher r, the differential acts like a generalized knight's move.