# Syllabus Math 505

## Math 505. Homological Algebra Instructor Syllabus

To understand an object, we break it into simpler components, for example, into a direct sum of irreducible objects. If an object B is irreducible, then a reasonable substitute is to include B in a short exact sequence 0 --> A --> B --> C --> 0; this is the starting point of homological algebra. This course covers:

1. Snake lemma, homology, long exact sequence in homology.
2. Projective and injective modules and resolutions.
3. Categories, functors and derived functors. Tor and Ext, local cohomology.
4. Group cohomology, bar resolution, homogeneous and inhomogeneous. Dimension shifting.
5. Spectral sequences, exact couples, convergence, techniques of computation, Serre spectral sequence. Grothendieck spectral sequence of composite functors.

Applications of spectral sequences are an important part of the course. Additional topics and applications will be chosen by the instructor according to his/her interest. Possibilities include: The equivalence of several definitions of Tor, the equivalence of definitions of Ext, the isomorphism of cellular and singular homology, Lyndon-Hochschild-Serre spectral sequence for group extensions, the spectral sequence generalizing the Mayer-Vietoris long exact sequence, Gysin sequence, Kunneth formula, universal coefficients theorem, homology of groups with coefficients in a chain complex, homology of amalgamations and HNN extensions of groups, Eilenberg-Moore spectral sequence for the homology groups of a pullback over a fibration, derived categories, étale cohomology.

Possible texts: A good main text would be Charles Weibel, An introduction to homological algebra. Additional material can be found in:

• Joseph J. Rotman, Introduction to Homological algebra. Second edition. 2009.
• Allen Hatcher, Spectral sequences in algebraic topology. Preliminary, three chapters of the book are available at http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html
• Spanier, Algebraic topology.
• Brown, Cohomology of groups.
• John McCleary, A user’s guide to spectral sequences.
• Saunders Mac Lane, Homology.