Page last update: September 3, 2008
MATH570 - HIGHER ALGEBRA I

FALL 2008

Lecturer: Dr. Eyal Goren
Office: BURN 1108
Office hours: Mon 14:00 - 15:00, Wed 15:00 - 16:00.
Lecture: MWF 10:30 - 11:30, BURN 920
Tutorial: M 11:30 - 13:00, BURN 1214
Quiz dates:   *  Monday, October 6, 11:30 - 13:00, BURN 1214.
*  TBA
*  TBA

NOTES

SYLLABUS
1. BASIC NOTIONS OF CATEGORY THEORY (4hrs)
• Categories and functors: the basic examples.
• Universal objects: products, coproducts, pullback and pushout, injective and projective limits.
• Adjoint functors. Equivalence of categories.
2. MODULE THEORY (14hrs)
• Recall of the basic theory.
• Modules over PIDs (Recall only).
• Tensor products.
• Projective, injective and flat modules and resolutions.
• The snake lemma and other trivial, but useful, diagrams.
• Derived functors. Remarks on Ext and Tor.
• Group Coholomology and applications (Hilbert's 90, forms).
3. SEMISIMPLE RINGS AND MODULES   (8hrs)
• Noetherian and Artinian rings and modules. Hilbert's theorem.
• Semisimple rings and modules - the basics.
• Nakayama's lemma and further study of Artinian modules.
• Jacobson's density theorem and the Artin-Weddrnburn theorem.
• The Brauer group.
4. REPRESENTATIONS OF FINITE GROUPS. (14 hrs)
• Definition and basic operations (sum, tensor product, dual, induction, symmetric and exterior products, symmetric square).
• Maschke's theorem and the structure of the group ring over an algebraically closed field.
• Character theory; behaviour under the basic operations; orthogonality of characters, class functions.
• Frobenius reciprocity.
• The representations of groups of small order and of dyhedral groups.
• Representations of S_n.
• Brauer's Theorem.

METHOD OF EVALUATION
There will be 3 quizzes during the semester: each worth 15% of the final grade.
There will be a final exam (in class) worth 55% of the final grade.
Exercises: exercises will be given frequently. Many of the easier proof and examples will be delegated to the exercises. The exercises will not be marked and will not carry a grade. Also, solutions will not be provided. All this for the simple reason of that you can find everything at this level at standard textbooks and the web. However, all the material in the exercises is included in the quizzes and the final. The exercise hour (to be fixed) will be devoted to discussing problematic exercises (per your requests) and I'd be happy to discuss your solutions with you during office hours.

TUTORIAL HOUR
There will be a weekly tutorial meeting to discuss exercises. The meeting will be Monday 11:30 - 13:00, BURN1214. As a rule, I will NOT solve the questions, but I will be willing to hear your solutions, or attempt at solution, and offer feedback. If I get irked enough with your approach, I'll solve the question.

TEXT BOOK
There is no official text book for the course, but the following books are recommended (the string in the end is the library call numbers. All books are on reserve in Schulich and Rosenthall):
Dummit and Foote: Abstract algebra QA162 D85 2004
Fulton and Harris: Representation theory QA 171 F85 1991
Jacobson: Basic algebra I, II  QA154.2 J32 1985
Lang: Algebra  QA154.3 L3 2002
Rotman: Introduction to homological algebra QA3 P8 v.65
Rotman: Advanced modern algebra QA 154.3 R68 2002
Serre: Linear representations of finite groups  QA171 S5313