determine Because E1 was exactly the homology with respect to d II, d II is zero on E1. Consequently, we get Using the other filtration gives us a different spectral sequence with a similar E2 term: What remains is to find a relationship between these two spectral sequences. It will turn out that as r increases, the two sequences will become similar enough to allow useful comparisons. Convergence, degeneration, and abutment Interpretation as a filtration of cycles and boundaries Let Er be a spectral sequence, starting with say r = 1. Then there is a sequence of subobjects such that ; indeed, recursively we let and let be so that are the kernel and the image of We then let and ; it is called the limiting term. (Of course, such need not exist in the category, but this is usually a non-issue since for example in the category of modules such limits exist or since in practice a spectral sequence one works with tends to degenerate; there are only finitely many inclusions in the sequence above.) Terms of convergence We say a spectral sequence converges weakly if there is a graded object with a filtration for every , and for every there exists an isomorphism . It converges to if the filtration is Hausdorff, i.e. . We write to mean that whenever p + q = n, converges to . We say that a spectral sequence abuts to if for every there is such that for all , . Then is the limiting term. The spectral sequence is regular or degenerates at if the differentials are zero for all . If in particular there is , such that the sheet is concentrated on a single row or a single column, then we say it collapses. 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. The spectral sequence of an unfiltered chain complex degenerates at the first sheet (see first example): since nothing happens after the zeroth sheet, the limiting sheet is the same as . The five-term exact sequence of a spectral sequence relates certain low-degree terms and E∞ terms. Examples of degeneration The spectral sequence of a filtered complex, continued Notice that we have a chain of inclusions: We can ask