Page last update: September 3, 2008
MATH570 - HIGHER ALGEBRA I
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
1. BASIC NOTIONS OF CATEGORY THEORY (4hrs)
2. MODULE THEORY (14hrs)
- Categories and functors: the basic examples.
- Universal objects: products, coproducts, pullback and pushout,
injective and projective limits.
- Adjoint functors. Equivalence of categories.
3. SEMISIMPLE RINGS AND MODULES (8hrs)
- 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).
4. REPRESENTATIONS OF FINITE GROUPS. (14 hrs)
- 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.
- 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
- Representations of S_n.
- Brauer's Theorem.
METHOD OF EVALUATION
There will be 3 quizzes during the semester: each worth 15% of the
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.
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
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.
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
algebra I, II
QA154.2 J32 1985
Lang: Algebra QA154.3
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