This needs to be merged with filtered chain complex
It’s easy to say what chain complex and homology mean (that is, these notions are definable); where things get tricky is, when calculating them, to figure out what the modules and differentials, kernels and images actually are. Sometimes there’s extra structure, e.g. a further hierarchy beyond the usual grading, that lets us figure these things out one layer at a time. Then we have to glue the layers back together, and that’s one place a spectral sequence is handy
(You may have noticed this isn’t the usual notation for functors. It’s traditional.)
The most frequent examples have , , or , with their usual total orderings; in this connection see spectral sequence of a filtered complex
Usually is a graded complex, with , and in this case we ask
(If you prefer cohomology differentials, read for .)
In the special case of a discrete totally-ordered filtration, one defines the associated graded complex with differential induced by ; again, if is graded, we have a bigraded complex with components and differential of bidegree .
Any book introducing spectral sequences.