Math 252: Introduction to affine algebraic geometry and commutative algebra (spring 2008)

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.