**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.