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The Spectrum of a Module Category
Henning Krause, University of Bielefeld, Germany
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Memoirs of the American Mathematical Society
2001; 125 pp; softcover
Volume: 149
ISBN-10: 0-8218-2618-2
ISBN-13: 978-0-8218-2618-8
List Price: US$57
Individual Members: US$34.20
Institutional Members: US$45.60
Order Code: MEMO/149/707
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These notes present an introduction into the spectrum of the category of modules over a ring. We discuss the general theory of pure-injective modules and concentrate on the isomorphism classes of indecomposable pure-injective modules which form the underlying set of this spectrum. The interplay between the spectrum and the category of finitely presented modules provides new insight into the geometrical and homological properties of the category of finitely presented modules. Various applications from representation theory of finite dimensional algebras are included.

Readership

Graduate students and research mathematicians interested in associative rings and algebras.

Table of Contents

  • Introduction
  • The functor category
  • Definable subcategories
  • Left approximations
  • duality
  • Ideals in the category of finitely presented modules
  • Endofinite modules
  • Krull-Gabriel dimension
  • The infinite radical
  • Functors between module categories
  • Tame algebras
  • Rings of definable scalars
  • Reflective definable subcategories
  • Sheaves
  • Tame hereditary algebras
  • Coherent rings
  • Appendix A. Locally coherent Grothendieck categories
  • Appendix B. Dimensions
  • Appendix C. Finitely presented functors and ideals
  • Bibliography
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