Notes and course
Notes on modules (only
Notes for the quick revision of modules
Old Quiz 1
Old Quiz 2
Old Quiz 3
Remarks on Ass 1 by Marc
Remarks on Ass 2 by Marc
on Ass 3 by Marc
Page last update:
October 23, 2009
MATH570 - HIGHER ALGEBRA I
Office: BURN 1108
Office hours: Mon 11:30 - 12:30, Fri 11:30 - 12:20.
Lecture: MWF 10:30 - 11:30, BURN 920
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
- The snake lemma and other trivial, but
BASIC NOTIONS OF CATEGORY THEORY
- Categories and functors: the basic
- Universal objects: products,
coproducts, pullback and pushout, injective and projective limits.
- Adjoint functors. Equivalence of
SEMISIMPLE RINGS AND MODULES
- Noetherian and Artinian rings and
modules. Hilbert's theorem.
- Semisimple rings and modules - the
- Nakayama's lemma and further study of
- Jacobson's density theorem and the
- The Brauer group.
REPRESENTATIONS OF FINITE GROUPS. (14 hrs)
- Definition and basic operations (sum,
tensor product, dual, induction, symmetric and exterior products,
- 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.
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
and Harris: Representation theory QA 171 F85 1991
Jacobson: Basic algebra I, II QA154.2 J32
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.