Markus Schmidmeier
Mathematical Sciences
Florida Atlantic University

# Modules

Welcome to my course on Modules (MAS 6396)! We meet MWF 11:00 - 11:50 p.m. in BU 403.

Topics

I am planning to cover several results about modules, in particular:

• The Jordan-Hölder Theorem,
• the Krull-Remak-Schmidt Theorem.
Let me explain what they are about.

Module theory is like lego playing. There are the simple modules, the bricks, they come in several different shapes and colors. We can stack them together to create nice models, modules. There are many beautiful possibilities, but watch out, some of those lego models or modules can get really big!

Now, it can happen (by chance or intention) that a lego model is dropped, then it will fall apart into its building blocks, the bricks. For modules, this is the set-up for the Jordan-Hölder Theorem: No matter how you drop it, the number of red bricks, green bricks, blue bricks etc. will always be the same!

Some models consist of many parts, if not, the model and the module will be called indecomposable. Suppose you keep putting bricks on top of an indecomposable model and remove bricks from the bottom. Then eventually, no matter how you proceed, none of the original bricks will be left in place! This is the famous Harada-Sai Lemma; it's only assumption is that the number of lego bricks involved in the process is bounded.

Regarding decomposable modules, one can ask about the multiplicites of indecomposable summands. As the modules get bigger, finding all the indecomposable summands of a certain type can get quite involved! This is the realm of the Krull-Remak-Schmidt Theorem...

Of course, you have been working with modules before, say when dealing with abelian groups (= Z-modules) or linear operators (= k[T]-modules). Also chains of subgroup embeddings, or systems consising of vector spaces and linear maps, are modules themselves! And they are important in pure mathematics (homological algebra, topology, commutative algebra, ...) and in applied mathematics (dynamical systems, control theory).

Objectives

• Gain familiarity with methods dealing with modules.
• Recognize modules as they come up in pure and applied mathematics.
• Practise presentations on advanced mathematical topics.

Prerequisites

Prerequisite is Linear Algebra 2 (MAS 4107), or higher. However, it is assumed that most students have taken the Introductory Abstract Algebra sequence (MAS 5311 and MAS 5312), and familiarity with algebraic structures (groups, rings, fields etc.) will be helpful throughout the course.

Textbook and and some literature

C. M. Ringel and J. Schröer, Representation Theory of Algebras: Modules, PDF. I plan to cover much of the material in the first three parts (pages 1-118).

The following graduate level textbooks are recommended for further study:

• F. W. Anderson, K. R. Fuller, Rings and categories of modules, Springer GTM 13 (1992).
• F. Kasch, Modules and rings, LMS Monographs 17 (1982).
• J. J. Rotman, Homological algebra, Academic Press (1979).
• M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, (1997).

Credit

Homework:  There will be weekly homework assignments, the 10 best will count for 40% of the grade.

Presentation:   Two presentations will count for 2×20% of the grade.

Written project:   One paper, written using a professional mathematics typesetting system, will count for 20% of the grade.

Final Exam:   The class will meet on December 7, at 10:30 a.m.; final exam credit consists of a presentation and the term paper.

Here you can find some general information about my courses.

Contact Me

Office hours:   MW, 2-4 p.m. in SE 230

Course Web Page:   http://math.fau.edu/schmidme/modules11.html

E-mail:   markus@math.fau.edu.