Commutative Algebra, Fall 2021

Lecturer:

  • Prof. Raphael Ponge.
  • E-mail: ponge[dot]math[at]icloud[dot]com.

Time and Location:

  • Tuesdays from 14:00-15:40, 3rd Teaching Building, Room 336.
  • Until I can come back to China the lectures will be online. Here is the Zoom link. Meeting ID: 986-0909-7168. Passcode: 075174.
  • The lectures are from Week 1 through Week 17. The last class is on December 21, 2021.

Main Reference:

  • Introduction to Commutative Algebra by M.F. Atiyah and I.G. MacDonald, Westview Press, 1969 (pdf).

Further Reading:

  • Exercises to Atiyah and Macdonald’s Introduction to Commutative Algebra by J.D.K. Carlson (pdf). These notes contain all solutions to the exercises in Atiyah-MacDonald’s book. They also contain proofs of results omitted or left to the reader in Atiyah-MacDonald’s book. Beware of some minor notation discrepancies with Atiyah-MacDonald’s book.
  • About Atiyah and MacDonald’s book by P.Y. Gaillard (pdf). These notes also contain solutions to the exercises in Atiyah-MacDonald. There are some useful additional comments on the main results in Atiyah-MacDonald’s book.
  • A Term of Commutative Algebra by A. Altman and S. Kleiman, Worldwide Center of Mathematics, 2017 (pdf). A modernized version of Atiyah-MacDonald’s book. Contains details that are not contained in MacDonald’s book or Carlson’s notes.

Grading Scheme:

  • You grade will be based on your scores on the quizzes and the Final Exam.
  • There will be 5 in-class quizzes throughout the semester.
  • The Final Exam will take place on December 21.

Quiz Info:

  • Each quiz will consists of 2~4 problems from the textbook (selected problems and/or exercises within the text).
  • Quiz 1 will take place on October 19 from 14:00-15:40 in Room 336 (3rd Teaching Building). It will cover Chapter 1 and Chapter 2. It will consist of 3 exercises from Atiyah-MacDonald’s book.
  • Quiz 2 & Quiz 3 will take place on November 16 from 14:00-15:40 in Room 336 (3rd Teaching Building). They will cover Chapter 3 and Chapter 4.
  • Quiz 3 is postponed to December 7.
  • Quiz 4 & Quiz 5 will take place on December 7 from 14:00-15:40 in Room 336 (3rd Teaching Building). They will cover Chapter 5, Chapter 6 and Chapter 7.

Selected Problems:

  • Chapter 1: #1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 22 (1st part only), 26, 28.
  • Chapter 2: #1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.
  • Chapter 3: # 1, 2, 3, 4, 5, 6, 7, 8, 9, 12[(i), (ii), (iii) only], 13, 14, 15.
  • Chapter 4: # 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16.
  • Chapter 5: #1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14.
  • Chapter 6: # 1, 2, 3, 4, 5, 6.
  • Chapter 7: # 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
  • Chapter 8: # 1, 2, 3, 6.

Schedule:

  • Sep. 7: Chapter 1 (Rings and Ideals) (slides).
  • Sep. 14: Chapter 1 (continued) + Chapter 2 (Modules) (slides).
  • Sep. 21: No classes, Mid-Autumn Festival.
  • Sep. 28: Chapter 2 (continued).
  • Oct. 5: No classes, National Day Holiday.
  • Oct. 12: Chapter 2 (continued).
  • Oct. 19: Quiz 1.
  • Oct. 26: Chapter 3 (Rings and Modules of Fractions) (slides).
  • Nov. 2: Chapter (continued) + Chapter 4 (Primary Decomposition) (slides).
  • Nov. 9: Chapter 4 (continued).
  • Nov. 16: Quiz 2.
  • Nov. 23: Chapter 5 (Integral Dependence and Valuations) (slides).
  • Nov. 25: Chapter 5 (continued) + Chapter 6 (Chain Conditions) (slides). Make-up lecture, from 18:30-20:10
  • Nov. 30: Chapter 6 (continued) + Chapter 7 (Noetherian Rings).
  • Dec. 7: Quiz 3, Quiz 4 & Quiz 5.
  • Dec. 14: Chapter 7 (continued) + Chapter 8 (Artin Rings).
  • Dec. 21: Final Exam.