'Et moi ¿.... si j'avait su comment en revenir. One service mathematics has rendered the human race. It has put common sense back je n'y serais point aUe.' it belongs. on the topmost shelf next Jules Verne where to the dusty canister labelled 'discarded non· The series is divergent: therefore we may be sense'. Eric T. Bell able to do something with it. o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

One, Preparatory.- 0. Algebras, Modules, Complexes.- I. Cohomology Groups and Problems Giving Rise to Them.- II. Tensor Product.- Two, Basic.- III. Homological Concepts (General Properties).- IV. Projectivity.- V. Resolutions and Dimensions.- VI. Multi-Operational Holomorphic Calculus on the Taylor Spectrum.- VII. Flatness and Amenability.- Appendix A. Paracompact topological spaces.- Appendix B. Invariant means on locally compact groups.- Postscript.- §1. Extensions and derivations.- §2. Normal cohomology and its expression in terms of Ext.- §4. An interpretation of amenability-according-to-Connes in terms of the diagonal and reduced bifunctionals.- §5. "General homological" background to amenability according to Connes.- §6. Central contractibility (= central separability) and central cohomology.- §7. Homological dimensions. Results of a general character and results connected with the geometry of Banach spaces.- §8. Homological dimensions (continued). Algebras of smooth functions and some radical algebras.- §10. Homological dimensions (concluded). Connections with the question of an analytic structure on the spectrum.- §11. Miscellaneous results about the homological invariants of operator algebras and their modules.- §12. Completely bounded cohomology and its applications.- §13. Weakly amenable Banach algebras and various conditions for "ordinary" and weak amenability.- §15. Some remarks about the development (and metamorphosis) of the problems of a multi-operator holomorphic calculus.- References.- Postscript references.- Index of terminology.- Index of notation and abbreviations.