Exact Couples and Spectral Sequences

In this chapter we develop the theory of spectral sequences: applications will be found in Section 9 and in Chapter IX. Our procedure will be to base the theory on the study of exact couples, but we do not claim, of course, that this is the unique way to present the theory; indeed, an alternative approach is to be found e.g. in [7]. Spectral sequences themselves frequently arise from filtered differential objects in an abelian category — for example, filtered chain complexes. In such cases it is naturally quite possible to pass directly from the filtered differential object to the spectral sequence without the intervention of the exact couple. However, we believe that the explicit study of the exact couple illuminates the nature of the spectral sequence and of its limit.