Course Announcement - Spring 2006

Math 250B: Commutative Algebra

Instructor: Bernd Sturmfels

Office hours: Wednesdays 8:30-11:00am and by appointment
Contact: bernd at math, 925 Evans, 642 4687

Teaching Assistant: Jason Morton

Time and Place: Tuesdays and Thursdays, 8:00-9:30am, 5 Evans Hall

Prerequisites: Math 250A or equivalent. Familiarity with algebraic geometry at the undergraduate level is helpful.
A good source for that undergraduate material is the book Ideals, Varieties and Algorithms by Cox, Little and O'Shea. Graduate students from outside the mathematics department are encouraged to talk to the instructor about their interests.

Course text: Gert-Martin Greuel and Gerhard Pfister:
A SINGULAR Introduction to Commutative Algebra, Springer Verlag, 2002.

Grading: The course grade will be based on weekly homeworks. The homework is due in class every Tuesday.
There is a strict no late homework policy. Students who wish to improve their standing in the course (e.g. to make up for missed homework) will be given an opportunity in April to write an additional term paper on a topic related to the course.

Homework: Assignments will be posted here. The problems refer to the course text.
(1) due January 24: 1.1.7, 1.1.12, 1.2.3, 1.2.4, 1.3.1, 1.3.9, 1.3.13, 1.3.16.
(2) due January 31: 1.4.1, 1.4.7, 1.4.8, 1.4.11, 1.5.2, 1.5.3, 1.5.4.
(3) due February 6: 1.6.3, 1.6.4, 1.7.4, 1.7.6, 1.7.10, 1.7.13, 1.7.15, 1.7.20.
(4) due February 13: 1.8.1, 1.8.2, 1.8.3, 1.8.4, 1.8.5, 1.8.6, 1.8.10, 1.8.12, 1.8.14.
(5) due February 20: 2.1.3, 2.1.9, 2.1.11, 2.1.14, 2.1.16, 2.1.18, 2.1.25, 2.1.26.
(6) due February 27: 2.1.20, 2.1.27, 2.2.1, 2.2.3, 2.2.4, 2.2.6, 2.2.7.
(7) due March 9 (Thursday): 2.4.2, 2.4.3, 2.4.5, 2.4.7, 2.4.9.
(8) due March 21: 5.1.1, 5.1.2, 5.1.3, 5.2.3, 5.2.5, 2.5.1, 2.5.3, 2.5.5, 2.5.7.
(9) due April 4: 2.7.4, 2.7.5, 2.7.7, 2.7.8, 2.8.1, 2.8.2, 2.8.5.
(10) due April 11: 3.1.1, 3.1.2, 3.1.4, 3.1.8, 3.2.1, 3.2.3, 3.2.9.
(11) due April 20 (Thursday): 3.3.5, 3.3.6, 3.3.7, 3.3.12, 3.4.1, 3.4.5, 3.4.7, 3.5.4.

Syllabus: This course is an introduction to commutative algebra including some computational and applied aspects. In parallel to our study of the mathematical concepts (local rings, primary decomposition, modules and their free resolutions, integral closure, normalization etc.), we shall learn how to use the computer algebra system SINGULAR. For the most part, I intend to follow the Greuel-Pfister book, but I will place more emphasis on Groebner bases in polynomial rings instead of standard bases in local rings. Also, I will stress the role of commutative algebra in methods for solving systems of polynomial equations.

In addition to [Greuel-Pfister], the following text books are recommended:

  • David Eisenbud: Commutative Algebra with a View toward Algebraic Geometry, Springer, 1995
  • Martin Kreuzer and Lorenzo Robbiano: Computational Commutative Algebra, Volumes 1 and 2, Springer, 2000 and 2005
  • Wolfram Decker and Christoph Lossen: Computing in Algebraic Geometry - A quick start using SINGULAR, Springer Verlag, Algorithms and Computation in Mathematics, Vol. 16, to appear in February 2006.