Markus Schmidmeier
Mathematical Sciences
Florida Atlantic UniversityWelcome to Homological Algebra, a.k.a. Abstract Algebra II, a.k.a. MAS 6314! We meet MW 5:00 - 6:20 PM in PS 109. Students who have taken the course MAS 6314 already may register for my course as a Directed Independent Study. Abstract Algebra II: Homological Algebra
Topics
An introduction to homological algebra, in particular for students with interests in algebra, geometry or topology. We will be covering:
- Modules and homomorphisms: short exact sequences, push-outs, pull-backs, snake lemma.
- Functors and modules they define: Hom and tensor functors; free, flat, projective and injective modules.
- Complexes: homotopy of morphisms, long exact homology sequence, projective resolutions, Ext and Tor.
- More about Ext: long exact sequence, vector space structure of first extension group.
- Homological dimensions and algebras they define.
- Related topics, e.g. an outlook to almost split sequences.
Prerequisites
Prerequisite for this course is some familiarity with modules, as for example obtained in Dr. Ford's course Rings and Modules (MAS 4906/MAS 6396), my Fall 2007 course, Abstract Algebra I (MAS 6313), or any similar course.
Textbook
C. M. Ringel and J. Schröer, Representation Theory of Algebras I and II, 2nd preliminary version, First Volume, and Second Volume. I plan to cover much of the material in Part IV (A-modules) in the first volume and Part I (Homological Algebra I) in the second volume.
Additional Literature
The classic by Rotman, Homological Algebra, is about to become available in a 2nd edition. I also list (in alphabetical order) several common textbooks that show the extent to which methods from homological algebra are used in algebra, geometry, topology.
- J. J. Rotman, Homological algebra, Academic Press (1st ed. 1979, 2nd ed. 12/2007).
- F. W. Anderson, K. R. Fuller, Rings and categories of modules, Springer GTM 13 (1992).
- M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, (1997).
- D. Eisenbud, Commutative Algebra with a view toward algebraic geometry, Springer GTM 150 (1995).
- J. R. Munkres, Elements of Algebraic Topology, Perseus Publishing (1984).
Credit
Homework: Weekly homework assignments will count for 40% of the grade.
Presentation: The presentation will count for 20% of the grade.
Final Exam: The final exam, a presentation on Monday 4/28 in SE 215, will count for 40% of the grade.
Contact Me
Office hours: MWF, 11-12 a.m. in SE 230
Course Web Page: http://www.math.fau.edu/schmidme/HomAlg08.html
E-mail: markus@math.fau.edu.
Last modified: by Markus Schmidmeier