Theory of spectral sequences. I

This is the first part of a paper on spectral sequences in an abelian category scr K. Here the author recalls Massey's definition of an exact couple, introduces a large variety of limit objects of such an exact couple by forming all different kinds of inverse limits varprojlim, resp. direct limits varinjlim, as well as the first derived limits, e.g., varinjlim{}^1. Then he introduces various kinds of convergences of a spectral sequence such as weak convergence, convergence, strong convergence and finally Boardman convergence. These types of convergence are associated with special exact sequences involving limit objects. In particular the author claims to extend a criterion of Boardman convergence to the case of abelian categories. In the final sections he deals with applications of his concepts to coherent (co)homology.