Homological Algebra
Homological algebra first arose as a language for describing topological properties of geometrical objects. Since then, it has expanded into subject on its own right, and its contemporary applications are many and diverse. A quick look at the Mathematics Subject Classification 2000 reveals application to Number Theory, Algebra and Differential Geometry, Lie Groups and Algebras, Finite Groups, Partial Differential Equations, Functional Analysis and Operator Theory. Therefore, the homological algebra methods must be in the toolbox of every mathematician.
The aim of this cause is to introduce the basic concepts and techniques in the language of categories and functors, and to present examples coming from various fields.
I. Simplicial sets
1. Triangulated spaces
2. Simplicial sets
3. Simplicial topological spaces and the Eilenberg-Zilber theorem
4. Sheaves
II. Homology and Cohomology
1. Complexes and morphisms of complexes
2. Coefficient systems
3. The long exact sequence
4. Homotopy
III. Examples
1. The Cech complex
2. The complex of singular chains
3. Homology and cohomology of groups
4. The de Rham complex
5. Homology and cohomology of Lie algebras
6. Hochschild (co)homology of algebras
7. Cyclic homology
8. The Koszul complex
IV. Categories and functors
1. Categories
2. Functors and natural transformations
3. Equivalences of categories
4. Adjoint functors
5. Additive and abelian categories
V. Derived functors
1. Injective modules and projective modules
2. Resolutions
3. Derived functors
4. Tor and Ext
5. Examples: (co)homology of sheaves, of groups, of Lie algebras, and of algebras
Bibliography
1. S.I. Gelfand and Yu.I. Manin - Methods of Homological Algebra, Springer-Verlag 1998
2. Ch. Weibel - An Introduction to Homological Algebra, Cambridge University Press, 1994
3. H. Cartan and S. Eilenberg - Homological Algebra, Princeton University Press, 1956
4. P. Hilton and U. Stammbach - A Course in Homological Algebra, Springer-Verlag 1971
5. S. Maclane - Homology, Springer-Verlag 1963
6. J. L. Loday - Cyclic Homology, Springer-Verlag 1992