Commutative Algebra

Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers , and p-adic integers. Commutative algebra is the main technical tool in the local study of schemes in algebraic geometry.

Classroom: 2-143

Time: Tu/Th 13-14:30

Email: cyxu [ at ] math.mit.edu

Textbook: Atiyah, M. F.; Macdonald, I. G. Introduction to commutative algebra.

Score: 40% Problem set 30% Midterm 30% Final report

Office hour:

ChenyangXu Thr 14:30-16:30, 2-468, Svetlana Makarova (murmuno@mit.edu) Mon 16:30-17:30, 2-231A (Oct. 15 14-15 and 16:30-17:30) or by appointment .

Tentative Schedule:

Nov. 6 **Midterm **Chapter 1-5 (The classes on Oct 31 and Nov 1 will be review)

Problem sets:

Chapter 1: 1-5, 10, 12-14, 15-21, 27, 28 (due Sep. 20 on class)

Chapter 2: 2, 3, 5, 7-10, 11(hard), 12 and Chapter 3: 1, 2, 5, 6 (due Oct. 4)

Chapter 3: 16, 17, 19, 21 and Chapter 4: 9, 10, 14, 15, 17, 18 (due Oct. 18)

Chapter 5: 7, 8, 12, 13, 14, 30, 31 (due Nov. 1)

Chapter 6: 1, 2, 5, 6; Chapter 7: 1, 2, 3, 5, 9, 14, 18; Chapter 8: 1, 4 (due Nov. 20)

Chapter 9: 1, 3, 8; Chapter 10: 3, 6, 7, 8, 9; Chapter 11 (optional): 1, 7 (due Dec. 4)

Some practice problems for the midterm (You don’t have to solve all of them!):

Besides the above P-set questions, Chapter 3: 3, 4, 7-9, 13-15, 18, 20, 22-25; Chapter 4: 1, 2, 4, 5, 7, 11-13, 16, 19; Chapter 5: 1-4, 9-11, 15, 26, 32, 35

Final Paper: The final will be writing an essay on one of the topics on my list. The topics will be posted on line on Dec. 5 6pm. It’s due on Dec. 12 6pm. The writing should include nontrivial examples, and show you really understand the materials.

See the Topics.

Other references: