Homological and homotopical aspects of Torsion theories / Apostolos Beligiannis, Idun Reiten.

Format:

Book

Author/Creator:

Beligiannis, Apostolos, 1969- author.

Reiten, Idun, 1942- author.

Series:

Memoirs of the American Mathematical Society ; Volume 188, Number 883.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 188, Number 883

Language:

English

Subjects (All):

Torsion theory (Algebra).

Algebra, Homological.

Homotopy theory.

Physical Description:

1 online resource (224 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2007.

Language Note:

English

Summary:

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 moregenerally (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 theselines. 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.

Contents:

""Contents""; ""Introduction""; ""Chapter I. Torsion Pairs in Abelian and Triangulated Categories""; ""1. Torsion Pairs in Abelian Categories""; ""2. Torsion Pairs in Triangulated Categories""; ""3. Tilting Torsion Pairs""; ""Chapter II. Torsion Pairs in Pretriangulated Categories""; ""1. Pretriangulated Categories""; ""2. Adjoints and Orthogonal Subcategories""; ""3. Torsion Pairs""; ""4. Torsion Pairs and Localization Sequences""; ""5. Lifting Torsion Pairs""; ""Chapter III. Compactly Generated Torsion Pairs in Triangulated Categories""; ""1. Torsion Pairs of Finite Type""

""2. Compactly Generated Torsion Pairs""""3. The Heart of a Compactly Generated Torsion Pair""; ""4. Torsion Pairs Induced by Tilting Objects""; ""Chapter IV. Hereditary Torsion Pairs in Triangulated Categories""; ""1. Hereditary Torsion Pairs""; ""2. Hereditary Torsion Pairs and Tilting""; ""3. Connections with the Homological Conjectures""; ""4. Concluding Remarks and Comments""; ""Chapter V. Torsion Pairs in Stable Categories""; ""1. A Description of Torsion Pairs""; ""2. Comparison of Subcategories""; ""3. Torsion and Cotorsion pairs""; ""4. Torsion Classes and Cohen-Macaulay Objects""

""5. Tilting Modules""""Chapter VI. Triangulated Torsion (-Free) Classes in Stable Categories""; ""1. Triangulated Subcategories""; ""2. Triangulated Torsion (-Free) Classes""; ""3. Cotorsion Triples""; ""4. Applications to Gorenstein Artin Algebras""; ""Chapter VII. Gorenstein Categories and ( Co) Torsion Pairs""; ""1. Dimensions and Cotorsion Pairs""; ""2. Gorenstein Categories, Cotorsion Pairs and Minimal Approximations""; ""3. The Gorenstein Extension of a Cohen-Macaulay Category""; ""4. Cohen-Macaulay Categories and ( Co) Torsion Pairs""

""Chapter VIII. Torsion Pairs and Closed Model Structures""""1. Preliminaries on Closed Model Categories""; ""2. Closed Model Structures and Approximation Sequences""; ""3. Cotorsion Pairs Arising from Closed Model Structures""; ""4. Closed Model Structures Arising from Cotorsion Pairs""; ""5. A Classification of ( Co) Torsion Pairs""; ""Chapter IX. ( Co) Torsion Pairs and Generalized Tate-Vogel Cohomology""; ""1. Hereditary Torsion Pairs and Homological Functors""; ""2. Torsion Pairs and Generalized Tate-Vogel ( Co-) Homology""

""3. Relative Homology and Generalized Tate-Vogel ( Co) Homology""""4. Cotorsion Triples and Complete Cohomology Theories""; ""Chapter X. Nakayama Categories and Cohen-Macaulay Cohomology""; ""1. Nakayama Categories and Cohen-Macaulay Objects""; ""2. ( Co) Torsion Pairs Induced by ( Co) Cohen-Macaulay Objects""; ""3. Cohen-Macaulay Cohomology""; ""Bibliography""; ""Index""

Notes:

"Volume 188, Number 883 (end of volume)."

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-0487-7