Relative Homological Algebra

This second volume deals with categories of complexes of modules and their applications. The material provides a good basis for those who want to go on to study derived categories. It starts with the basic definitions and properties of complexes and then moves to more advanced topics such as cotorsion pairs and triplets in categories of complexes. As in the first volume, exercises will be provided both for practice and as suggestions for further research.

This second volume deals with the relative homological algebra of complexes of modules and their applications. It is a concrete and easy introduction to the kind of homological algebra which has been developed in the last 50 years. The book serves as a bridge between the traditional texts on homological algebra and more advanced topics such as triangulated and derived categories or model category structures. It addresses to readers who have had a course in classical homological algebra, as well as to researchers.

Autorentext

Edgar E. Enochs, University of Kentucky, Lexington, USA; Overtoun M. G. Jenda, Auburn University, Alabama, USA.

Zusammenfassung

"The authors' presentation is original, condensed and carefully written. Even in case the authors' would be right with their claim that the results "are probably well known" it will be an important help for a person who will become familiar with these details to find them in such a compact and clearly presented way." Zentralblatt für Mathematik

Inhalt
DedicationPreface Chapter I: Complexes of Modules1. Definitions and basic constructions2. Complexes formed from Modules3. Free Complexes4. Projective and Injective Complexes Chapter II: Short Exact Sequences of Complexe1. The groups Extn(C, D)2. The Group Ext1(C, D)3. The Snake Lemma for Complexes4. Mapping Cones Chapter III: The Category K(R-Mod)1. Homotopies2. The category K(R-Mod)3. Split short exact sequences4. The complexes Hom(C, D)5. The Koszul Complex Chapter IV: Cotorsion Pairs and Triplets in C(R-Mod)1. Cotorsion Pairs2. Cotorsion triplets3. The Dold triplet4. More on cotorsion pairs and triplets Chapter V: Adjoint Functors1. Adjoint functors Chapter VI: Model Structures1. Model Structures on C(R-Mod) Chapter VII: Creating Cotorsion Pairs1. Creating Cotorsion pairs in C(R-Mod) in a Termwise Manner2. The Hill lemma3. More cotorsion pairs4. More Hovey pairs Chapter VIII: Minimal Complexes1. Minimal resolutions2. Decomposing a complex Chapter IX: Cartan and Eilenberg Resolutions1. Cartan-Eilenberg Projective Complexes2. Cartan and Eilenberg Projective resolutions3. C - E injective complexes and resolutions4. Cartan and Eilenberg Balance Bibliographical NotesReferencesIndex