The goal is to lift information (aka cohomology) from the associated graded complex to that of the original complex. The first approximation is . The cycles are what lands in a lower layer (or upper index plus one) of the filtration. Thus it is natural to let be the elements that drop levels down the filtration, and be the (decreasing) intersection of all of them. The boundary are the image of elements that comes from levels above the filtration, and be the increasing union of all of them. Let . Note that has a differential, but what is the grading? Note that the filtration number increase by but the grading on the complex only increase by . So to make things more symmetric we define so then . This grading is natural if is the total complex of some double complex.
A more general way in which spectral sequence arise is via exact couples. Some spectral sequences cannot be constructed from filtered complexes, see this discussion. The disadvantage is that it can be the case that our objects of study are not explicitly filtered or do not come from a filtered differential object. In this section we present another general algebraic setting, exact couples, in which spectral sequences arise. The ease of definition of the spectral sequence and its applicability make this approach very attractive. Unlike the case of a filtered differential graded module, however, the target of the spectral sequence coming from an exact couple may be difficult to identify.
It is easy to see for double complexes concentrated in upper left quadrant, stablize as long as . For Serre’s spectral sequence we use pullback of a cellular filtration of the base and get a filtration of . The theorem is that there is a filtration on with associated graded . But we still need to solve some extension problem over if we want to determine .
Example:
For the path space fibration ( simply connected) Since the path space is contractible, we know what the spectral sequence is converging to, and we immediately get .
Reference: