I have been reading this book with the main purpose of introducing myself to spectral sequences. Although so far I've been enjoying it quite a lot, I've had a problem I cannot handle:
In theorem 14.6 (page 160) they introduce a graded complex and a given filtration. Afterwards they say that the filtration has to induce a filtration in each one of the components of the grading (that they call dimensions). I have no idea why this has to be true, actually if the given grading is the one used to define the cohomology of K, then this makes no sense at all (the only way I see this making sense is if we force the grading to be invariant under the differential, a requisition that would be a bit weird, to say the least, since in the major part of the cases the differential is a 1-level graded map in the complex). Can someone help me figure this out? Thanks!