In Chapter 2, we find recipes for the construction of spectral sequences. To develop these ideas further we need to clarify the relationship between a spectral sequence and its target; this is the goal of Chapter 3. To achieve this goal, it is necessary to introduce more refined ideas of convergence. These ideas require a discussion of limits and colimits of modules and the definition of a morphism between spectral sequences with which one can express the relevant theorems of comparison. In the case of a filtered differential graded module, conditions on the filtration guarantee that the associated spectral sequence converges uniquely to its target. The case of an exact couple is more subtle and we develop it after a discussion of some associated limits.

We express convergence results as comparison theorems that answer the questions: If two spectral sequences are isomorphic via a morphism of spectral sequences, then how do the targets of the spectral sequences compare? Need they be isomorphic? We end the chapter with some constructions and Zeeman's comparison theorem that reveals how special circumstances lead to powerful conclusions.

On convergence

Theorem 2.6 tells us that a filtered differential graded module, (A, d, F), determines a spectral sequence and, if the filtration is bounded, then the spectral sequence determines H(A, d) (up to extension problems). We want to remove the restrictive hypothesis of a bounded filtration and still retain convergence to a uniquely determined target.