Spectral sequence Homological algebra

fix abelian category, such category of modules on ring. spectral sequence choice of nonnegative integer r0 , collection of 3 sequences:

a doubly graded spectral sequence has tremendous amount of data keep track of, there common visualization technique makes structure of spectral sequence clearer. have 3 indices, r, p, , q. each r, imagine have sheet of graph paper. on sheet, take p horizontal direction , q vertical direction. @ each lattice point have object




e

r


p
,
q




{\displaystyle e_{r}^{p,q}}

.

it common n = p + q natural index in spectral sequence. n runs diagonally, northwest southeast, across each sheet. in homological case, differentials have bidegree (−r, r − 1), decrease n one. in cohomological case, n increased one. when r zero, differential moves objects 1 space down or up. similar differential on chain complex. when r one, differential moves objects 1 space left or right. when r two, differential moves objects knight s move in chess. higher r, differential acts generalized knight s move.