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Course Announcement - Spring 2006

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Math 250B: Commutative Algebra

**Office hours:**
Wednesdays 8:30-11:00am and by appointment

**Contact:** bernd at math, 925 Evans, 642 4687

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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.

** 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.