Math 252: Introduction to affine algebraic geometry (spring 2007)

Instructor: Chad Schoen

Time and Place: MWF 8:45 - 9:35, room 205 Physics Building.

Instructor's office: room 191 Physics Building.

E-mail: schoen@math.duke.edu

Telephone: 660-2813

Office Hours: 9:50 - 10:50 Monday and Friday.

Target audience: This is a basic, introductory course in commutative algebra and algebraic geometry suitable for anyone who has taken math 251.

Course Description : The course will introduce algebraic geometry which is a core subject in mathematics. Very loosely speaking algebraic geometry is that branch of mathematics which studies the geometry of figures defined by polynomial equations in several variables. The simplest examples of such figures are familiar from high school (lines, planes, conic sections, caustics, cardoid, four leaved rose,...). We will quickly see that this list barely scratchs the surface. Furthermore, we will quickly find that algebraic geometry is based on commutative algebra and that progress in geometry requires us to develope basic themes in commutative algebra including extension and contraction of ideals, finite and integral extensions of rings, localization, completion, and dimension theory. Commutative algebra is also the basis for much of algebraic number theory. However this course will be strongly biased towards algebraic geometry. To keep things from getting too complicated we will focus on affine algebraic varieties. The course should prepare participants for Math 273 in which quasi-projective algebraic varieties are defined and studied. Math 252 is an essential course for students considering working in algebraic geometry or a related algebraic field. It is a prerequisite for all subsequent courses in algebraic geometry and for some courses in several complex variables and algebraic number theory.

Text: The course will not follow a text as closely as Math 251 did. A larger fraction of the homework will come from handouts. An excellent reference for commutative algebra is: Introduction to Commutative Algebra, by Atiyah and Macdonald

Homework: Weekly homework assignments.

Grading : Grading will be based on homework and any projects and exams. A final exam is possible. A mid-term is unlikely.

Links to some pictures of algebraic sets Algebraic curves: (Joel Robert's webpage)
Plane curves (many of which are algebraic, but some of which are not):
Algebraic curves and surfaces:
A gallery of algebraic surfaces: (Bruce Hunt)
Wolf Barth's movies of families of surfaces which change as their equations are modified:
(Click on small frames at the left): movies

Homework assignments:

1. For Friday, January 19.
Pictures: Check out the links to the pictures of algebraic sets given above.
Reading: Please read each of the following. Sorry about the length of this assignment.
Most of the passages are short and several are mostly review.
(i) Zorn's Lemma (which one may take as an axiom of set theory). Artin, Algebra page 588.
(ii) Basic properties of rings and ideals (mostly review): Atiyah and MacDonald Chapter 1.
(Skip the section on the nilradical and Jacobson radical. Aslo skip Propositions 1.11
through 1.16.)
(iii) Review basic properties of finitely generated modules over a Noetherian ring:
Artin, Chapter 12, 5.13 - 5.17.
Atiyah and MacDonald, Chapter 6 through Proposition 6.2.
(iv) The Hilbert basis theorem: Artin, Chapter 12, 5.18 - 5.25. Atiyah-MacDonald Chapter 7
through 7.7 skipping 7.3 and 7.4.
Optional reading:
(v) The correspondence between ideals and algebraic sets (called affine varieties in CLO).
Cox, Little and O'Shea, Chapter 1 sections 2 and 4.

Please read sections 1.1, 1.2, 1.3 of the handouts.
Non-written exercises:
1. Let $I_1, I_2\subset R$ be ideals in a commutative ring.
Recall the definitions of the ideals $I_1+I_2, I_1\cap I_2, I_1I_2$.
Handout 1.2: Exercises 1, 2, 3, 5.
Written exercises:
Handout 1.2: 4,6.

2. For Friday, January 26
Reading:
The proof of the weak form of the Nullstellensatz given in class followed
Atiyah-MacDonald, 7.8-7.10.
A different proof of the Nullstellensatz. Artin Chapter 10 sections 7 and 8 (through 8.7).
Exercises 1.3 1-3, 1.4 1-10.

3. For Firday, February 2
Reading. If you have time, Aityah-MacDonald Chapt 2 through 2.9.
These sections are mostly review.
We won't really hit modules for at least another week, so this is
not urgent.
Exercises 1.5 1-10 and 1.6 1-2.
There is quite a bit here. Do what you can. No need to write up anything.
We'll discuss them on Friday.

3. For Firday, February 9
Read Atiyah-MacDonald 2.10.
Exercises on handout.

4. For Firday, February 16
Read in Atiyah-MacDonald Chapter 2: Tensor product of modules through
the end of the chapter.
Exercises from handouts: 2.1Ex2, 2.1Ex3, 2.0Ex2, 2.0Ex3(i), 2.0Ex5, 1.7Ex4,5,8,9,10.

5. For Firday, February 23
Read Atiyah-MacDonald Chapter 3. (You may skip 3.14 and 3.15.)
Atiyah_MacDonald Chapter 3 which one could look at: Ex 4,5,7,8,9, 12(ignore hint).
Exercises from handouts: 2.2Ex1,2,3,6,7 2.3Ex2,3,5,7,8,9,10
Each person is responsible for 5 exercises from the list above.

6. For Firday, March 2
Read Atiyah-MacDonald: The first two sections of Chapter 5. Integral dependence and going up.

7. For Firday, March 9
Discuss problems from the handout.

8. For Firday, March 23
Read Artin, section 13.8.

9. For Friday, March 30
Reading in Atiyah-Macdonald:
Nakayama's Lemma: 2.3-2.8; A somewhat different approach
to 2.4 is given in the notes Theorem and Cor. 3.1, 3.1Ex11.
Integral closure: 5.12-5.13. Discrete valuation rings is a section in Chapter 9.
This pertains more to homework for next week.

For Friday, April 6
Reading in Atiyah-Macdonald:
Discrete valuation rings is a section in Chapter 9.
Nilpotent elements in a ring. 1.7-1.8.
Going down: 5.14-5.16.

For Friday, April 13
Reading in Atiyah-Macdonald:
Graded rings: 10.7.
Local dimension=dimension for irreducible alg. sets: 11.25 - 11.27.
Going down from a different point of view: 3.16, 5.14-5.16
Completions: Chapter 10 through 10.6.

For Friday, April 20
Completions 10.8 - 10.23.
Local dimension theory: Chapter 11.