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