User:Marty/UCSC Math 203 Fall 2008

=Syllabus=

Mathematics Covered
The following is the official course guide description of the class:
 * Topics include tensor product of modules over rings, projective modules and injective modules, Jacobson radical, Weederburns' theorem, category theory, Noetherian rings, Artinian rings, affine varieties, projective varieties, Hilbert's Nullstellensatz, prime spectrum, Zariski topology, discrete valuation rings, and Dedekind domains.

We will follow these topics pretty closely, though in a different order. This class should prepare students for the techniques involved in algebraic geometry and algebraic number theory. In particular, by understanding rings and their spectra, students will gain a geometric intuition for algebra, and an algebraic intuition for geometry.

Text, Exercises, and Lecture
We will be approaching these topics in the following way: we will be going through much of the book "Introduction to Commutative Algebra", by Atiyah and MacDonald, hereafter referred to as "A-M". This text covers almost all of the above topics, though we might digress occasionally to discuss some noncommutative topics. The real meat of the book is in the exercises. We will attempt to go through Chapters 1,2,3,5,6,7,8,9 of this book. Students will be required to work on a large number of exercises (essentially all of the exercises) from each of these chapters. Students are expected to work in small groups in order to facilitate this work.

Half of classtime will be devoted to lecture. Lecture will provide background, examples, and motivation, to accompany the technical material. The other half of classtime will be devoted to a discussion of the exercises. Students will be expected to present their solutions, and attempts at solutions. Every enrolled student will be required to speak on his or her solutions, at various times.

The exercises in A-M vary greatly in difficulty: students are not expected to solve every problem, but an earnest attempt is expected. Many problems rely on the results of other problems. It is nearly impossible to go through the problems in A-M, while skipping some along the way.

Grades
Grades will be determined by the quality of their problem solutions, as evidenced by written work and presentation. A take-home final will be given as well, covering the material throughout the course.

Administration
All details of the course will be given on the SlugMath wiki. The URL for this course is. Students should create an account on the wiki by going to the wiki, and creating an account. Students should not edit the wiki, outside of discussion pages, without permission. Students can discuss problems and their solutions, using the discussion pages for each week. Discussion and collaboration is highly encouraged.

For additional information and advice, Marty can be contacted easily by e-mail at weissman AT ucsc DOT edu. Office hours are readily available by appointment.

=Week-by-week=

Week 1

 * Days: MW, Sept. 29, Oct. 1.
 * Admin: Introductions. Syllabus.  About the book and exercises.  Grading.  The Wiki.
 * Math: Gel'fand duality.  Why algebra and geometry together?  Spectra.  Rings of functions.  Rings of operators.  Rings of numbers.  An arsenal of examples.
 * Lecture notes: Download [[Image:Math203Week1.pdf]]

Week 2

 * Days: MW, Oct. 6,8.
 * Problems: Students lecture on A-M, Ch. 1, Problems 1-14.
 * Math: Point-set topology.  Synthetic and incidence geometry.  Non-Hausdorff topology.  Closed and generic points.  Rings of functions.  Cutting out subspaces.  Zero-divisors.  The need for nonreduced geometry.
 * Lecture notes: Download [[Image:Math203Week2.pdf]]

Week 3

 * Days: MW, Oct. 13,15.
 * Problems; Students lecture on A-M, Ch. 1, Problems 15-28.
 * Math: The abelian category of modules.  Differentials and the algebraic de-Rham sequence.  Lots of examples of tensor products.  Extension of scalars.  Torsion.  Partial exactness and derived functors.
 * Lecture notes: Download [[Image:M203Notes3.pdf]].  These were originally notes from my Math 202 class, but it seems like a good idea to put them here.  They introduce the "abelian category" of modules.

Week 4

 * Days: MW, Oct. 20,22.
 * Problems: Students lecture on A-M, Ch. 2, Problems 1-28.
 * Math: Localization in geometry and algebra.  Universal properties.  The sheaf concept.  The Mittag-Leffler problem.

Week 5

 * Days: MW, Oct. 27,29.
 * Problems: Students lecture on A-M, Ch. 3, Problems 1-30.
 * Math: Lecture notes on derivations

Week 6
An intermission from Atiyah-Macdonald!
 * Days: MW, Nov. 3,5.
 * Math: Some homological/categorical lecture notes.

Week 7

 * Days: MW, Nov. 10,12.
 * Problems: Students lecture on A-M, Ch. 5, Problems 1-15.
 * Math: Here are some notes:  [[Image:Math203Week6.pdf]].  Even better, Ravi has some excellent notes on algebraic geometry.  Look at the notes from Class 19 and 20, for material relevant to Chapter 5 of Atiyah-Macdonald.

Week 8

 * Days: MW, Nov. 17,19.
 * Problems: Students lecture on A-M, Ch. 5, Problems 16-35.
 * Math: Here are some lecture notes on norms.

Week 9

 * Days: MW, Nov. 24, (26).
 * Problems: Students lecture on A-M, Ch. 6.

Week 10

 * Days: MW, Dec. 1,3.
 * Problems: Students lecture on A-M, Ch. 7
 * Math: Here are some [[Image:Math203Week10.pdf‎|lecture notes on Grobner bases]].  Notes by the inventor can be found here.  However, these notes are difficult (for me) to follow since I am not sure what ordering the author chooses for monomials.