Cornell Math - Math 632 (SP01)

Math 632 — Spring 2001
Algebra

Prerequisite: Math 631 or equivalent.

This course should be accessible to beginning graduate students and will cover material that will be essential to anyone interested in ring theory, homological algebra, representation theory, or K-theory and should be of value for students of algebraic topology and number theory as well as to others.

The main content of this course is to study the ``simplest'' rings (those of dimension 0), to characterize them, to relate this in various ways to other concepts in algebra, and to give useful applications. Our approach to the study of semisimple rings is homological rather than ring-theoretic as this leads to results more quickly and gives a cleaner, easier to understand approach.

The philosophy of the course will be to ``learn by doing'' with a corresponding set of homework assignments.

Course Text: Farb & Dennis, Noncommutative Algebra, Graduate Texts in Mathematics, vol. 144, Springer-Verlag, 1993. (Review of the text.)

We will cover material in the following chapters:

0. Background Material

This chapter covers the prerequisites for the course. Although this will not be covered separately, relevant parts will be blended into the general presentation.

1. Semisimple Modules & Rings and the Wedderburn Structure Theorem

We cover the basics of semisimple modules and rings, the Wedderburn Structure Theorem, several equivalent definitions of semisimplicity, a structure theorem for simple artinian rings, and Maschke's Theorem.

2. The Jacobson Radical

Various definitions of radical are given and connected with the concept of semisimplicity. Nakayama's Lemma, local rings, and the radical of a module are also covered.

3. Central Simple Algebras

We discuss extension of scalars and semisimplicity, prove the Skolem-Noether and Double-Centralizer theorems, derive the classical theorem of Wedderburn that finite division rings are fields, and give Frobenius' classification of the central division algebras over the real numbers.

4. The Brauer Group

The Brauer group and relative Brauer group are defined. The general study of $Br(k)$ is reduced to that of studying $Br(K/k)$ for galois extensions $K/k$. Group cohomology is introduced, and an explicit description of the Brauer group is given.

6. Burnside's Theorem and Representations of Finite Groups

As an application of our earlier work, we study the representations of a finite group over the complex numbers, introduce characters and prove the orthogonality relations. Burnside's famous theorem that every finite group of order $p^aq^b$ is solvable then follows easily.

Suggested references:

N. J. Divinsky, Rings and Radicals, Mathematical Expositions 14, Allen and Unwin, London, 1965.

I. N. Herstein, Noncommutative Rings, Carus Mathematical Monographs, No. 15, 1968.

N. Jacobson, Basic Algebra I, II, W.H. Freeman and Company, San Francisco, 1980.

J. P. Jans, Rings and Homology, Holt, Rinehart and Winston, 1964.

L. Rowen, Ring Theory, Vols. I and II, Academic Press, 1988.