Notes and course material

Notes on modules (only partially proof-read)

Notes for the quick revision of modules

Exercises

 
Old Quiz 1
Old Quiz 2
Old Quiz 3
Old Final

 

Remarks on Ass 1 by Marc

Remarks on Ass 2 by Marc
Remarks on Ass 3 by Marc

                                                                  Page last update: October 23, 2009
MATH570 - HIGHER ALGEBRA I
FALL 2009

Lecturer: Prof. Eyal Goren

Course Assistant: Marc Masdeu
Office: BURN 1108
Office hours: Mon 11:30 - 12:30, Fri 11:30 - 12:20.
Lecture: MWF 10:30 - 11:30, BURN 920
Midterm: TBA


                                                                                                        
SYLLABUS
1. MODULE THEORY

  • Recall of the basic theory.
  • Modules over PIDs (Recall only).
  • Localizaton of rings and modules (mostly through exercises)
  • Tensor products.
  • Projective, injective and flat modules and resolutions.
  • The snake lemma and other trivial, but useful, diagrams.

2. BASIC NOTIONS OF CATEGORY THEORY

  • 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  

  • 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.


METHOD OF EVALUATION

  • There will be a midterm worth 25% of the final grade.
  • The assignments will be marked (selected exercises only) and worth 20% of the grade.
  • There will be a final exam (in class) worth 55% of the final grade.


TEXT BOOKS
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

Academic integrity: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

Submitting work. In accord with McGill University’s Charter of Students’ Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

Syllabus and Grade Calculation. In the event of extraordinary circumstances beyond the University’s control, the content and/or evaluation scheme in this course is subject to change.