Purity, Spectra and Localisation

Description

The central aim of this book is to understand modules and the categories they form through associated structures and dimensions, which reflect the complexity of these, and similar, categories. The structures and dimensions considered arise particularly through the application of model-theoretic and functor-category ideas and methods. Purity and associated notions are central, localization is an ever-present theme and various types of spectrum play organizing roles. This book presents a unified, coherent account of material which is often presented from very different viewpoints and clarifies the relationships between these various approaches Content: Preface Introduction Part I. Modules: 1. Pp conditions 2. Purity 3. Pp pairs and definable subcategories 4. Pp-types and pure-injectivity 5. The Ziegler spectrum 6. Rings of definable scalars 7. m-dimension and width 8. Examples 9. Ideals in mod-R A. Model theory Part II. Functors: 10. Finitely presented functors 11. Serre subcategories and localisation 12. The Ziegler spectrum and injective functors 13. Dimensions 14. The Zariski spectrum and the sheaf of definable scalars 15. Artin algebras 16. Finitely accessible and presentable additive categories 17. Spectra of triangulated categories B. Languages for definable categories C. A model theory/functor category dictionary Part III. Definable categories: 18. Definable categories and interpretation functors D. Model theory of modules: an update E. Glossary Main examples Bibliography Index.