Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Homological Algebra

Welcome to my course on Homological Algebra (MAS 6396/ MAT 4930)! We meet Tuesdays and Thursdays 9:30 - 10:50 a.m. in BU 102.

Modules are ubiquitous in modern mathematics, and by now you have met them on several occasions, for example disguised as abelian groups (Z-modules), as vector spaces (k-modules), in Intro Abstract Algebra (as modules over Euclidean domains), in group theory (as representations) etc.

Working with modules is like lego playing. There are the simple modules, the bricks, they come in several different shapes and colors. Many modules are semisimple (that is, direct sums of simples), we can think of them just as a bunch of lego bricks. This is boring.

The fun begins when you put the bricks together. This is what homological algebra is all about. We can stick many bricks together, and we can stick them together in many ways. What we get is called an extension and such things can get really fancy.

Topics

An introduction to homological algebra, in particular for students with interests in algebra, geometry or topology. We will be covering:

• Constructions of modules and homomorphisms: short exact sequences, push-outs, pull-backs, snake lemma.
• Functors, and modules defined by functors: 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.

Objectives

• Recognize the use of categories and functors in various areas in mathematics.
• Use concepts like projectivity, resolutions and Ext-groups to study systems of vector spaces and linear maps.
• Investigate complexes using the long exact homology sequence.
• Practise written and oral presentations on advanced topics in mathematics.

Textbook

C. M. Ringel and J. Schröer, Representation Theory of Algebras, 2nd preliminary version. I plan to cover selected sections in Part 1 (Categories and modules, examples), Part 3 (Modules over rings), Part 4 (Projective modules) and Part 5 (Homological Algebra I).

The textbook by Rotman, Homological Algebra, is recommended for this course. It is now available in the 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...

• 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).
• I. Assem, D. Simson, A. Skowroński, Elements of representation theory of associative algebras: 1, LMSST 65 (2006).
• 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:  There will be biweekly homework assignments, the 6 best will count for 60% of the grade.

Presentation:   Two presentations will count for each 20% of the grade.

Final Exam Day:   The last class meeting will be on Thursday, April 27, during 7:45 - 10:15.

Here you can find some general information about my courses.

Contact Me

Office hours:   TR 4-5:30 p.m. in SE 272

Course Web Page:   http://math.fau.edu/markus/homalg17.html

E-mail:   markus@math.fau.edu.