## McGill University

# Department of Mathematics & Statistics

# Higher Algebra II

# 189-571B

## Detailed Syllabus

** 0. Wednesday January 3 to Friday January 5**

**Text**: Jacobson, Basic Algebra I, Chapter 7.1.

Overview of the course.
Algebras over a field. Definition and examples of
associative algebras.

** 1. Monday January 8 to Friday January 12 **

**Text**: Jacobson, Basic Algebra I, Chapter 7.2-7.4.

Algebras over a field, cont'd. Exterior algebras and determinants.
Matrix representations of associative algebras.

Assignment 1

Remarks on the assignment.

** 2. Monday January 15 to Friday January 19 **

**Text**: Jacobson, Basic Algebra I, Chapter 7.7.

Associative division algebras. Theorems of Frobenius and Wedderburn.

Assignment 2

Remarks on the assignment.

** 3. Monday January 22 to Friday January 26 **

**Text**: Jacobson, Basic Algebra II, Chapter 4.1-4.2.

Structure theory of (non-commutative) rings. Primitivity,
Jacobson radical.

Assignment 3

Remarks on the assignment.

** 4. Monday January 29 to Friday February 2 **

**Text**: Jacobson, Basic Algebra II, Chapter 4.3.

Density Theorems.

Assignment 4

Remarks on the assignment.

** 5. Monday February 5 to Friday February 9 **

**Text**: Jacobson, Basic Algebra II, Chapter 4.4-4.5.

Artinian rings. Structure theory of algebras.

Assignment 5

Remarks on the assignment.

** 6. Monday February 12 to Friday February 16 **

**Text**: Jacobson, Basic Algebra II, Chapter 4.6-4.7.

Finite dimensional
central simple algebras.
The Brauer Group.

Assignment 6

Remarks on the assignment.

**Monday February 19 to Monday February 23**.

Study Break.

** 7. Monday February 26 to Friday March 2 **

Some review.

Assignment 7

Remarks on the assignment.

The *Midterm* will be held this week, on Wednesday, February 28,
at 11:30 AM, in room 1205.

** 8. Monday March 5 to Friday March 9 **

**Text**: Humphreys, Introduction to Lie algebras and
representation theory, chapter I.

**Suppplementary Text**: Fulton and Harris, Representation theory,
lectures 7 and 8.

A brief introduction to Lie algebras.

Assignment 8

Remarks on the assignment.

** 9. Monday March 12 to Friday March 16 **

Basic Algebra II, Chapters 6.1-6.4.

Complexes, homology, long exact homology sequence.

Assignment 9

Remarks on the assignment.

** 10. Monday March 19 to Friday March 23 **

Basic Algebra II, Chapter 6.5-6.6.

Homological algebra: resolutions, derived functors. Examples.

Assignment 10

Remarks on the assignment.

** 11. Monday March 26 to Friday March 30 **

**Text**: Jacobson, Basic Algebra II, Chapter 6.7-6.8.

Homological algebra: Ext and Tor.

Assignment 11

Remarks on the assignment.

** 12. Monday April 2 to Friday April 6 **

**Text**: Jacobson, Basic Algebra II, Chapter 6.9-6.10.

Homological algebra: More on Ext, and the rudiments of the
Cohomology of groups.

Practice Final