Theory of Spectral Sequences of Exact Couples: Applications To Countably And Transfinitely Filtered Modules

Abstract

This thesis has two parts. In the first part we start from an arbitrary exact couple of R-modules and describe completely how the E-infinity terms of the associated spectral sequence relate to adjacent filtration stages of the universal (co-)augmenting objects of the exact couple. This advances earlier work, notably that of Boardman. In the second part we use these insights to develop a framework which permits spectral sequence methods to gain information about suitably transfinitely filtered objects. We offer several applications of this method:

We use Serre's idea of working relative to a class of modules while passing through the pages of the spectral sequence associated to an exact couple and we spell out conditions under which the filtration stages of countably or transfinitely filtered modules stay within such a class. We extend Zeeman's comparison technique of spectral sequences to apply to a map between countably or transfinitely filtered modules. Finally, we develop a general setting of reverse engineering information about finite pages in a spectral sequence from information about the universally filtered objects of the underlying exact couple.