Here are the course materials for Commutative Algebra, taught at Leiden University in Spring 2014.

**Lecturer:** Owen Biesel

**Time and Place:** 1:45 to 3:30 in Snellius room 401.

**Office Hours:** 2:00 to 3:00 on Wednesdays in Snellius room 206d.

**Email:** bieselod@math.leidenuniv.nl (no longer valid; use the contact page instead)

**Recommended Reading:** “Commutative Algebra with a View Toward Algebraic Geometry” by David Eisenbud.

### Course Outline (so far):

**February 3:**Noetherian rings and modules. (pdf) Homework 1

Recommended reading: sections 1.1-1.9 of Eisenbud.

**February 10:**Localization and universal properties. (pdf) Homework 2

Recommended reading: section 2.1 of Eisenbud.

**February 17:**Tensor Products and Hom. (pdf) Homework 3

Recommended reading: sections 2.2-2.3 of Eisenbud.

**February 24:**Supports of Modules. (pdf) Homework 4

Recommended reading: none this week.

**March 3:**Associated Primes. (pdf) Homework 5

Recommended reading: sections 3.1-3.2 of Eisenbud.

**March 10:**No class.**March 17:**Primary Decomposition. (pdf) Homework 6

Recommended reading: sections 3.3-3.9 of Eisenbud.

**March 24:**Integrality and the Nullstellensatz. (pdf) Homework 7

Recommended reading: chapter 4 of Eisenbud.

**March 31:**Completions and the Artin-Rees Lemma. (pdf) Homework 8

Recommended reading: sections 5.0, 5.2, and chapter 7 of Eisenbud.

**April 7:**Flat Modules and Algebras. (pdf) Homework 9

Recommended reading: sections 6.0-6.4 of Eisenbud.

**April 14:**Dimension (Zero) and Artinian Rings. (pdf) Homework 10

Recommended reading: section 2.4 and chapters 8-9 of Eisenbud.

**April 21:**No class**April 28:**Going Up and Down. (pdf) Homework 11

Recommended reading: sections 4.4, 10.0, and 10.2 of Eisenbud.

**May 5:**No class**May 12:**Associated Graded Rings. (pdf) Homework 12

Recommended reading: sections 5.1, 5.3-5.4, and 7.5 of Eisenbud.

**May 19:**The Fundamental Theorem of Dimension Theory. (pdf) Homework 13

Recommended reading: section 1.9, section 10.1, and chapter 12 of Eisenbud.

### Announcements (in reverse chronological order):

June 30: The resit exam will be July 1, from 10am to 1pm in **Snellius room 174**. Good luck!

June 18: Congratulations on your work on the final exam! I have posted the exam above, together with my comments. I have also posted your grades for the final exam, along with the weighted average with your homework scores as it stands now. If you choose to resit the exam on July 1, only the maximum of your two exam scores will count.

June 5: Due to a conflict with the Algebraic Geometry final exam, I have also reserved Snellius 401 for the hours of 10am-noon on Tuesday, and will be holding a review session then as well.

June 3: I will be holding a review session on Tuesday, June 10, from 3pm to 5pm in Room 401 of Snellius. Come with questions and we’ll try to solve them together.

May 31: Final exam information:

- The exam is on June 12, from 2pm to 5pm in room 412 of Snellius.
- The exam will consist of two parts. The main part will have three homework-style problems with multiple parts. The second, smaller part will consist of five multiple-choice questions about the material covered in class and the homework.
- For the homework-style questions, you may use results from lecture without re-proving them, but I may ask you to solve problems very similar or identical to problems on the homework.
- For the multiple-choice questions, no proofs of your answers will be required.
- Tips for studying: Review the lecture notes and make sure you understand where and how the hypotheses are used in each theorem. Review the homework and make sure you understand how to solve each problem, including the optional ones. Old exams are also a good source of practice problems, though of course some questions refer to topics we did not cover.
**Clarifications:**You will not be allowed to bring any written materials with you, either lecture notes or old homeworks. Also, if a result was stated in class but proved in the homework, you can still cite it without proof on the exam*unless*I am asking you to re-prove it.- If you desire to retake the exam, the resit is scheduled for July 1, from 10am to 1pm. The structure of the resit exam will be identical to the original, though of course the questions will be different.

Best of luck studying! Please email me if you have any questions.

May 28: Homework 12 grades are up. Soon I will post another announcement about what to expect (and how to study for) the final exam.

May 17: Several people have had questions about the degree-preserving aspect of part 1a on Homework 12. Perhaps it will help to clarify that the initial form of an element r and the initial form of its image f(r) can have different degrees; this is not a problem. In other words, you’ll build a map from the associated graded ring of R to the associated graded ring of R’; you can make a square of functions using it, the original ring homomorphism R -> R’, and the initial form maps. This square will not (in general) commute.

May 7: Grades for Homework 10 have been posted. It has also been pointed out to me that those of you who aren’t attending the lectures have only the recommended Eisenbud readings as a clue to what we covered in class each week, and sometimes I include material that isn’t contained there. To remedy this situation, I will be posting my personal notes for each lecture over the next few weeks; my hope is that notes for all the lectures will be available in time to help with completing the remaining homework assignments and reviewing for the final exam.

April 29: Office hours are canceled this week (due to the Bachelorseminarium) and next week (as I will be out of town). I will remain in email contact in case of questions.

April 18: There was an error in Exercise 4 of Homework 10 as well; the module with the desired properties was M/Z, not M itself. I have corrected the statement and put in a more precise hint for part a.

April 14: There was an error in the hint for Exercise 4b of Homework 9. I have corrected the hint, but it makes the exercise a little harder than I intended, and I apologize. I will be working with the graders to make sure everyone’s submission is graded fairly despite my mistake. Homework 10 and grades for Homework 8 are now online.

April 1: Homework 8 is posted; it appears quite long but much of the text is hints. The website migration is complete, and all old links should redirect to the new site.

March 7: Homework 4 grades are posted.

March 5: There was an error in the statement of HW5, Exercise 1: the module M should have been A/(xy,y^2), not A/(x^2,xy).

March 4: Homework 5 is due on March 17, not March 10. Homework will always be due on the day of class following its assignment.

March 3: Homework 3 grades are posted.

February 19: Homework 2 grades are posted. All future homework assignments will list the weight of each problem.

February 12: Homework 1 grades are posted.

February 10: Message from Graders

- Grader contact information:
- Mima: mima_ck89@hotmail.it – Snellius room 252
- Iuliana: iuliana.ciocanea@gmail.com – Snellius room 234

- On Monday, 10th February, we asked the students present at the lecture to fill in their name, university and email address in a list. If you are not on that list, please email us with that information, so we can put you on it.
- When you email us, always email us BOTH! Please also include the “magic” word “CommA” in the subject line, so we can sort our emails faster, e.g. CommA HW2, CommA Question.
- If you have a question you want to ask us, you are welcome to come by our offices. However, it might be better to just email us and we will get back to you with an answer.
- Every week, after the grading is done, you will be able to check your grade on the course website (i.e. here!).

February 3: Grading

**20%: Homework.**Each assignment is posted on Tuesday and due in class at the beginning of the following lecture. You are encouraged to work together to solve problems, but each of you should submit your own work.**80%: Written Final Exam.**The exam is scheduled for 2:00-5:00 on Thursday, June 12. The resit time is 10:00-1:00 on July 1.