Homological Algebra

Annotation

A semester course introducing the basic constructions and techniques of homological algebra used in algebraic topology, algebraic geometry, and forming the basis of a number of geometric methods in mathematical physics.

The topics covered include chain complexes and differential graded al- gebras, quasi-isomorphisms, projective and injective modules, resolu- tions, homological dimension, Tor and Ext functors, regular sequences and Cohen–Macaulay rings, bicomplexes and filtered complexes, spec- tral sequences, A_∞-morphisms.

Prerequisites: a basic course in algebra (groups, rings, modules, vector spaces), basic concepts of topology (continuous maps, homotopy).

Course plan

1. Algebras and modules, chain complexes, differential graded al- gebras, homology.
2. Chain homotopies, quais-isomorphisms, long exact sequences.
3. Free, projective and injective modules, resolutions.
4. Tor and Ext.
5. Examples of resolutions: minimal resolution of graded modules, Koszul resolution, bar an cobar constructions.
6. Systems of parameters, regular sequences and Cohen–Macaulay algebras.
7. Projective dimension and depth of modules.
8. Multiplications:  Eilenberg–Zilber and Ku¨nneth theorems.
9. Formality and Massey products.
10. Spectral sequences of a filtered complex and bicomplex.
11. A_∞-morphisms, extended functoriality of Tor, the Eilenberg– Moore spectral sequence.

Literature

1. Bruns, Winfried; Herzog, Ju¨rgen.  Cohen–Macaulay  Rings.  Re- vised edition. Cambridge Studies in Adv. Math., 39, Cam- bridge Univ. Press, Cambridge, 1993.
2. Eisenbud, David. Commutative Algebra with a View Toward Al- gebraic Geometry. Graduate Texts in Mathematics, 150. Springer, New York, 1995.
3. Hatcher, Allen. Algebraic Topology. Cambridge Univ. Press, Cambridge, 2002
4. Mac Lane, Saunders. Homology. Springer, Berlin, 1963.
5. Weibel, Charles. An Introduction to Homological Algebra. Cam- bridge University Press, Cambridge, 1994.