Theory of Spectral Sequences. I

A theory of spectral sequences in an arbitrary abelian category is described. Dual techniques for constructing spectral sequences are presented (the sequences obtained are the same but with different limit objects in general). The following objects are studied: limit objects of exact projective and injective couples, limit objects of spectral sequences, and different types of convergence (very weak and weak, strong and complete, and conditional convergence in the sense of Boardman). Theorems on complete composition rows for right-half-plane, left-half-plane, and whole-plane spectral sequences are considered. This theory is applied to (co)chain complexes, coherent (co)homology and (co)homotopy, as well as to Bockstein exact couples.