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Homological and Homotopical Aspects of Torsion Theories
Apostolos Beligiannis, University of Ioannina, Greece, and Idun Reiten, Norwegian University of Science and Technology, Trondheim, Norway
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Memoirs of the American Mathematical Society
2007; 207 pp; softcover
Volume: 188
ISBN-10: 0-8218-3996-9
ISBN-13: 978-0-8218-3996-6
List Price: US$74
Individual Members: US$44.40
Institutional Members: US$59.20
Order Code: MEMO/188/883
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In this paper the authors investigate homological and homotopical aspects of a concept of torsion which is general enough to cover torsion and cotorsion pairs in abelian categories, \(t\)-structures and recollements in triangulated categories, and torsion pairs in stable categories. The proper conceptual framework for this study is the general setting of pretriangulated categories, an omnipresent class of additive categories which includes abelian, triangulated, stable, and more generally (homotopy categories of) closed model categories in the sense of Quillen, as special cases.

The main focus of their study is on the investigation of the strong connections and the interplay between (co)torsion pairs and tilting theory in abelian, triangulated and stable categories on one hand, and universal cohomology theories induced by torsion pairs on the other hand. These new universal cohomology theories provide a natural generalization of the Tate-Vogel (co)homology theory. The authors also study the connections between torsion theories and closed model structures, which allow them to classify all cotorsion pairs in an abelian category and all torsion pairs in a stable category, in homotopical terms. For instance they obtain a classification of (co)tilting modules along these lines. Finally they give torsion theoretic applications to the structure of Gorenstein and Cohen-Macaulay categories, which provide a natural generalization of Gorenstein and Cohen-Macaulay rings.

Table of Contents

  • Introduction
  • Torsion pairs in abelian and triangulated categories
  • Torsion pairs in pretriangulated categories
  • Compactly generated torsion pairs in triangulated categories
  • Hereditary torsion pairs in triangulated categories
  • Torsion pairs in stable categories
  • Triangulated torsion(-free) classes in stable categories
  • Gorenstein categories and (co)torsion pairs
  • Torsion pairs and closed model structures
  • (Co)torsion pairs and generalized Tate-Vogel cohomology
  • Nakayama categories and Cohen-Macaulay cohomology
  • Bibliography
  • Index
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