Spectral Sequence - Convergence, Degeneration, and Abutment

Convergence, Degeneration, and Abutment

In the elementary example that we began with, the sheets of the spectral sequence were constant once r was at least 1. In that setup it makes sense to take the limit of the sequence of sheets: Since nothing happens after the zeroth sheet, the limiting sheet E_∞ is the same as E₁.

In more general situations, limiting sheets often exist and are always interesting. They are one of the most powerful aspects of spectral sequences. We say that a spectral sequence converges to or abuts to if there is an r(p, q) such that for all r ≥ r(p, q), the differentials and are zero. This forces to be isomorphic to for large r. In symbols, we write:

The p indicates the filtration index. It is very common to write the term on the left-hand side of the abutment, because this is the most useful term of most spectral sequences.

In most spectral sequences, the term is not naturally a doubly graded object. Instead, there are usually terms which come with a natural filtration . In these cases, we set . We define convergence in the same way as before, but we write

to mean that whenever p + q = n, converges to .

The simplest situation in which we can determine convergence is when the spectral sequences degenerates. We say that the spectral sequences degenerates at sheet r if, for any s ≥ r, the differential d_s is zero. This implies that E_r ≅ E_r+1 ≅ E_r+2 ≅ ... In particular, it implies that E_r is isomorphic to E_∞. This is what happened in our first, trivial example of an unfiltered chain complex: The spectral sequence degenerated at the first sheet. In general, if a doubly graded spectral sequence is zero outside of a horizontal or vertical strip, the spectral sequence will degenerate, because later differentials will always go to or from an object not in the strip.

The spectral sequence also converges if vanishes for all p less than some p₀ and for all q less than some q₀. If p₀ and q₀ can be chosen to be zero, this is called a first-quadrant spectral sequence. This sequence converges because each object is a fixed distance away from the edge of the non-zero region. Consequently, for a fixed p and q, the differential on later sheets always maps from or to the zero object; more visually, the differential leaves the quadrant where the terms are nonzero. The spectral sequence need not degenerate, however, because the differential maps might not all be zero at once. Similarly, the spectral sequence also converges if vanishes for all p greater than some p₀ and for all q greater than some q₀.

The five-term exact sequence of a spectral sequence relates certain low-degree terms and E_∞ terms.