**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**: 9:45 - 10:45 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 or
anyone whose knowledge of rings, ideals, modules, and Galois theory is roughly
at the level of someone who has completed math 251. Math 252 is an essential course for individuals who are interested in algebraic geometry, commutative algebra, algebraic number theory, singularity theory, or the local theory of several complex variables.

** Course Description **: The goal of the course sequence
Math 252-Math 273 is to introduce
students to algebraic geometry. Math 252 concentrates on affine
algebraic varieties. It introduces the commutative algebra necessary
to work with these objects. Basic topics in commutative algebra
which will be covered include extension and contraction of ideals,
finite and integral extensions of rings, localization, completion,
and dimension theory. Geometric topics which will be covered
include decomposition into irreducible components, dimension
computations, morphisms between varieties, quotients by group actions, singularities, tangent spaces, obstructions to embedding.
The sequel, Math 273, builds on these foundations.
With the help fo sheaves (discussed in Math 272)
one patches affine algebraic varieties together to create
quasi-projective varieties, which are the basic objects
of study in algebraic geometry.

**Text**: During the course of the semester we will read a
large fraction of the excellent textbook, Introduction to Commutative
Algebra, by Atiyah and Macdonald. However, it would not be
correct to say that the course will follow the textbook. Indeed
the course will develope algebraic geometry from the internal logic
of the geometry, while the text developes commutative algebra from
the internal logic of the algebra. When results in commutative
algebra are needed for the geometric developement (which is roughly
every class period), then we will pick up the necessary commutative
algebra from Atiyah and Macdonald.

** Grading **: Grading will be based on homework and any
projects and exams. A final exam is possible.

**Homework**:

First assignment: Please read:

(i) Handout: Sections 1.1 and 1.2.

(ii) Basic properties of rings and ideals (mostly review):

Atiyah and MacDonald Chapter 1. (Skip the section on the nilradical and Jacobson radical. Also skip Propositions 1.11 through 1.16.)

(iii) Review basic properties of finitely generated modules over a Noetherian ring:

Atiyah and MacDonald, Chapter 6 through Proposition 6.2.

(Alternate source: Artin, Algebra, Chapter 12, 5.13 - 5.17.)

(iv) The Hilbert basis theorem: Atiyah-MacDonald Chapter 7
through 7.7 skipping 7.3 and 7.4.

(Alternate source: Artin, Chapter 12, 5.18 - 5.25.)

Please prepare for discussion the following problems from the handout:

1.1 Exercises: 1

1.2 Exercises: 1, 2-4, 7.

For Monday, January 28

Please read: Atiyah-MacDonald, 7.8-7.10.

Please read: Handout sections 1.3 and 1.4.

Please prepare for discussion the following problems from the handout:

1.3 exercises 1, 2.

1.4 exercises 1-10.

For Monday, February 4

Please read: Handout sections 1.5 and 1.6

Please prepare for discussion the following problems from the handout:

1.4 exercises 11-15

1.5 exercises 1-4, 6

1.6 exercise 1,4

For Monday, February 11

Please read: Handout section 1.7

Please prepare for discussion the following problems from the handout:

1.5 exercises 6, 7, 13, 14, 15

1.6 exercises 2-4

1.7 exercises 2,3,4

For Monday, February 18

Please read: Handout section 1.8 and 2.2.

Please prepare for discussion the following problems from the handout:

1.8 exercises 2-6.

2.2 exercises 1-4.

For Monday, February 25

Please read: Handout sections 2.1 and 2.3

Please read: Atiyah-Macdonald Chapter 2 through 2.20.

Please prepare for discussion the following problems from the handout:

2.1 exercises 1-3, 4(ii), 8

2.3 exercises 1-6

For Monday, March 3

Please read: Handout section 2.4, 2.5

Please read: Atiyah-Macdonald Chapter 3

Please read: Handout sections 2.4, 2.5, 2.6, 2.7.

Please prepare for discussion the following problems from the handout:

2.4 exercises 2-5

2.5 exercises 3(iii)-(v), 4(i), 5.

For Monday, March 10

Have a nice spring break!

For Wednesday, March 19

Please read: Handout sections 3.5 and 4.1

Please read: Atiyah-Macdonald Chapter 5 through 5.11

Please prepare for discussion the following problems from the handout:

2.6 exercises 1,2

2.7 exercises 2, 3, 4

3.5 exercise 2

4.1 exercises 1,2,3,4,5,6,7

For Monday, March 24

Please read: Handout sections 4.2, 4.3

Please read: Atiyah-Macdonald pages 30-31

Please prepare for discussion the following problems from the handout:

4.1 exercises 8, 9, 10

4.2 exercises 1-6.

For Monday, March 31

Please read: Handout sections 4.4, 4.5

Please prepare for discussion the following problems from the handout:

4.4 exercises 1-4. 5(i-iv)

4.5 exercises 1-3, 5.

For Monday, April 7

Please read: Handout sections 4.6, 6.1, 6.2

Please read propositions 5.12 and 5.13 in Atiyah and Macdonald

The first three pages of chapter 9 in Atiyah and Macdonald

(stop at Dedekind domains) are recommended reading

Please prepare for discussion the following problems from the handout:

4.6 exercises 2, 3, 5

6.1 exercises 1,5,6

6.2 exercises 2,3

For Monday, April 14

Please read: Handout sections 6.4 and 5.2

Please read about filtrations and graded rings and modules

in Atiyah-Macdonald through Proposition 10.11.

Also read the first paragraph on the associated graded ring

before Proposition 10.22.

Please prepare for discussion the following problems from the handout:

6.1 exercises 4,6

5.2 exercises 1-3, 5-7

6.2 exercise 1

Notes: Going down is treated in Atiyah-Macdonald Theorem 5.16.

The proof given is different than the one in class. There is

not a lot of motivation for the argument. Finiteness of integral

closure for finitely generated algebras over a field is treated

in Proposition 5.17. The argument is based on non-degeneracy of

the trace form, which is not a standard topic in Math 251. This

is a standard result in algebraic number theory and will be

covered in any course on this topic.

For Monday, April 21

Please read Atiyah-Macdonald Chapter 11.

Please read: Handout sections 7.2 and 7.3.

Please prepare for discussion the following problems from the handout:

5.2 exercises 3, 8

7.2 exercises 1-4

7.3 exercises 1-4

Final assignment

Please read Atiyah-Macdonald Chapter 10.