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Purity, Spectra and Localisation


Part of Encyclopedia of Mathematics and its Applications

  • Date Published: June 2009
  • availability: Available
  • format: Hardback
  • isbn: 9780521873086

£ 185.00

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About the Authors
  • 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.

    • Inclusion of relevant background material makes this suitable as an introduction for postgraduate students and researchers from other areas
    • Compiles a wide range of methods and results, some of which are previously unpublished, into one single, accessible source
    • Contains an extensive index, a detailed table of contents and thorough internal referencing
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    Reviews & endorsements

    'This book is an account of a fruitful interaction between algebra, mathematical logic, and category theory. … provides relevant background material and a wealth of illustrative examples. An extensive index and thorough referencing also make this book an ideal, comprehensive reference.' L'Enseignement Mathématique

    'This very interesting book is written so that it will be useful for graduate students and experienced researchers. It is worth to be mentioned that Mike Prest introduced in this book some very useful appendixes which come to help the reader to become familiar with the language of model theory. The book contains a lot of illuminating examples which are also very helpful for the reader.' Zentralblatt MATH

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    Product details

    • Date Published: June 2009
    • format: Hardback
    • isbn: 9780521873086
    • length: 798 pages
    • dimensions: 240 x 164 x 45 mm
    • weight: 1.35kg
    • availability: Available
  • Table of Contents

    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

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    Purity, Spectra and Localisation

    Mike Prest

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  • Author

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

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