The Spectral Sequence of A Filtered Complex, Continued
Notice that we have a chain of inclusions:
We can ask what happens if we define
is a natural candidate for the abutment of this spectral sequence. Convergence is not automatic, but happens in many cases. In particular, if the filtration is finite and consists of exactly r nontrivial steps, then the spectral sequence degenerates after the r'th sheet. Convergence also occurs if the complex and the filtration are both bounded below or both bounded above.
To describe the abutment of our spectral sequence in more detail, notice that we have the formulas:
To see what this implies for recall that we assumed that the filtration was separated. This implies that as r increases, the kernels shrink, until we are left with . For, recall that we assumed that the filtration was exhaustive. This implies that as r increases, the images grow until we reach . We conclude
- ,
that is, the abutment of the spectral sequence is the p'th graded part of the p+q'th homology of C. If our spectral sequence converges, then we conclude that:
Read more about this topic: Spectral Sequence, Examples of Degeneration
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