Purity, Spectra and Localisation

by Mike Prest

Overview

It is possible to associate a topological space to the category of modules over any ring. This space, the Ziegler spectrum, is based on the indecomposable pure-injective modules. Although the Ziegler spectrum arose within the model theory of modules and plays a central role in that subject, this book concentrates specifically on its algebraic aspects and uses. The central aim 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, localisation is an ever-present theme and various types of spectrum play organising 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.

Product Details

ISBN-13:	9781139637541
Publisher:	Cambridge University Press
Publication date:	06/04/2009
Series:	Encyclopedia of Mathematics and its Applications , #121
Sold by:	Barnes & Noble
Format:	eBook
File size:	32 MB
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About the Author

Mike Prest is Professor of Pure Mathematics at the University of Manchester.

Table of Contents

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.

Editorial Reviews

"Prest's book is encyclopedic in scope. It will certainly serve as the bible of model theory of modules for years to come. However, it is also a book of examples: the interested reader can start with a very concrete example and watch how, as the book develops, this example treads on more and more abstract notions."
Gena Puninski, Mathematical Reviews

From the Publisher