Homology

1. Front Matter

   Pages I-X

   PDF

2. Introduction

   • Saunders Mac Lane

   Pages 1-8

3. Modules, Diagrams, and Functors

   • Saunders Mac Lane

   Pages 8-34

4. Homology of Complexes

   • Saunders Mac Lane

   Pages 34-63

5. Extensions and Resolutions

   • Saunders Mac Lane

   Pages 63-103

6. Cohomology of Groups

   • Saunders Mac Lane

   Pages 103-138

7. Tensor and Torsion Products

   • Saunders Mac Lane

   Pages 138-172

8. Types of Algebras

   • Saunders Mac Lane

   Pages 173-200

9. Dimension

   • Saunders Mac Lane

   Pages 200-220

10. Products

    • Saunders Mac Lane

    Pages 220-248

11. Relative Homological Algebra

    • Saunders Mac Lane

    Pages 248-279

12. Cohomology of Algebraic Systems

    • Saunders Mac Lane

    Pages 280-318

13. Spectral Sequences

    • Saunders Mac Lane

    Pages 318-358

14. Derived Functors

    • Saunders Mac Lane

    Pages 358-403

15. Back Matter

    Pages 404-422

    PDF

In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL­ LING. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL ElLENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. Both REINHOLD BAER and JOSEF SCHMID read and commented on my entire manuscript; their advice has led to many improvements. ANDERS KOCK and JACQUES RIGUET have read the entire galley proof and caught many slips and obscurities. Among the others whose sug­ gestions have served me well, I note FRANK ADAMS, LOUIS AUSLANDER, WILFRED COCKCROFT, ALBRECHT DOLD, GEOFFREY HORROCKS, FRIED­ RICH KASCH, JOHANN LEICHT, ARUNAS LIULEVICIUS, JOHN MOORE, DIE­ TER PUPPE, JOSEPH YAO, and a number of my current students at the University of Chicago - not to m~ntion the auditors of my lectures at Chicago, Heidelberg, Bonn, Frankfurt, and Aarhus. My wife, DOROTHY, has cheerfully typed more versions of more chapters than she would like to count. Messrs.

• Abelian group
• Factor
• algebra
• auditor
• cohomology
• collaboration
• commutative ring
• development
• group
• homological algebra
• homology
• proof
• sets
• topology
• university

• Department of Mathematics, University of Chicago, Chicago, USA

  Saunders Mac Lane

Biography of Saunders Mac Lane

Saunders Mac Lane was born on August 4, 1909 in Connecticut. He studied at Yale University and then at the University of Chicago and at Göttingen, where he received the D.Phil. in 1934. He has tought at Harvard, Cornell and the University of Chicago.

Mac Lane's initial research was in logic and in algebraic number theory (valuation theory). With Samuel Eilenberg he published fifteen papers on algebraic topology. A number of them involved the initial steps in the cohomology of groups and in other aspects of homological algebra - as well as the discovery of category theory. His famous and undergraduate textbook Survey of modern algebra, written jointly with G. Birkhoff, has remained in print for over 50 years. Mac Lane is also the author of several other highly successful books.

Policies and ethics