MATH 571: Higher Algebra 2
Winter 2011
Course description

Instructor

Course information

There will be a mid-term and final exam, as well as weekly exercises. The final grading will be based on the exercises (20 %), mid-term exam (20 %) and final exam (60 %).

Exercises

Exercises are usually to be handed in on Mondays.

Textbooks

Recommended:

1. Dommit, David S; Foote, Richard M.; Abstract Algebra. Wiley; 3 edition (July 14, 2003), 944 pp., ISBN: 978-0471433347

Other:

1. Eisenbud, David; Commutative Algebra with a View Toward Algebraic Geometry Springer-Verlag, New York, 1995, ISBN: 978-0387942698
2. Lang, Serge; Algebra. Revised third edition. Graduate Texts in Mathematics, 211. Springer-Verlag, New York, 2002. xvi+914 pp. ISBN: 0-387-95385-X
3. Jacobson, Nathan; Basic algebra. II. Second edition. W. H. Freeman and Company, New York, 1989. xviii+686 pp. ISBN: 0-7167-1933-9
4. Atiyah, M. F.; Macdonald, I. G.; Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.

Syllabus

Rings, Part I.

1. Integral extensions and the going-up and going-down theorems.
2. Noether's normalization lemma, and Hilbert's Nullstellensatz.
3. Noetherian and Artinian rings.
4. Hilbert's basis theorem.

Modules, Part II.

1. Tensor products.
2. Projektive modules.
3. Injective modules.
4. Flat modules.

Rings, Part II.

1. The Jacobson radical.
2. Nakayama's lemma.
3. Semisimple rings and modules.
4. Jacobson's density theorem and the Artin-Wedderburn theorem.

Categories, Part III.

1. Exact functors.
2. Adjoint functors.

Groups, Part II.

1. Linear representations of groups.
2. Maschke's theorem.
3. Characters. Orthogonality of characters. Frobenius Reciprocity.
4. Representations of nilpotent groups.
5. Representations of the symmetric group.
6. Representations of GL_2(F), for F a finite field.

- if time is left - Homological algebra.

1. The snake lemma and the 5 term lemma.
2. Projective and injective resolutions and derived functors.
3. The derived functors Tor and Ext.
4. Homotopy and independence on the resolution.

Notice

McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offenses under the Code of Student Conduct and Disciplinary Procedures (see McGill web page on Academic Integrity for more information).