**Memoirs of the American Mathematical Society**

2007;
207 pp;
Softcover

MSC: Primary 18;
Secondary 16; 20; 55

Print ISBN: 978-0-8218-3996-6

Product Code: MEMO/188/883

List Price: $78.00

Individual Member Price: $46.80

**Electronic ISBN: 978-1-4704-0487-1
Product Code: MEMO/188/883.E**

List Price: $78.00

Individual Member Price: $46.80

# Homological and Homotopical Aspects of Torsion Theories

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*Apostolos Beligiannis; Idun Reiten*

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

# Table of Contents

## Homological and Homotopical Aspects of Torsion Theories

- Contents v6 free
- Introduction 110 free
- Chapter I. Torsion Pairs in Abelian and Triangulated Categories 817 free
- Chapter II. Torsion Pairs in Pretriangulated Categories 2231
- Chapter III. Compactly Generated Torsion Pairs in Triangulated Categories 4352
- Chapter IV. Hereditary Torsion Pairs in Triangulated Categories 6069
- Chapter V. Torsion Pairs in Stable Categories 7988
- Chapter VI. Triangulated Torsion (-Free) Classes in Stable Categories 102111
- Chapter VII. Gorenstein Categories and ( Co) Torsion Pairs 117126
- Chapter VIII. Torsion Pairs and Closed Model Structures 132141
- Chapter IX. ( Co) Torsion Pairs and Generalized Tate-Vogel Cohomology 163172
- Chapter X. Nakayama Categories and Cohen-Macaulay Cohomology 186195
- Bibliography 200209
- Index 204213 free