MA8202 Commutative Algebra (Spring 2020)

Exam:

  • The exams will be oral exams held on Zoom.
  • You should now have received a confirmation of your desired exam time. You will receive a Zoom link closer to the exam.
  • The exams will cover all material covered in class, in the textbook, and on the assigned exercises.
  • The exams will primarily take place during week 20 (11-15 May). If you have multiple exams that week, please contact me individually to inquire about alternative exam times later in the month. In either case, you must still schedule your exam no later than Thursday 30 April.

News:

  • 4.5: I will hold an open office hour for questions from 9-10am on Thursday 7.May on Zoom. This is not intended to be a lecture but rather a time that you can come and go with specific questions. You can join the zoom meeting with Meeting ID 658-2793-8809 (the password can be found on blackboard, or you can email me directly). If you have additional questions, feel free to email me questions throughout the week.
  • 20.4: Please fill out this brief opinion poll regarding examination dates: https://forms.gle/vR8YDzkZ3pykt2ak9
  • 1.4: Torgeir Aambø has written up a proof of Hilbert's nullstellensatz which uses Noether normalization which I might suggest you read after you have completed chapter 7 exercise #14.
  • 25.3: Please complete this 1-minute feedback form to evaluate how you feel the adjustments are during this time where we are not able to meet in person: https://forms.gle/EFLEcnSHuVJ4vN9EA
  • 14.3: Update posted regarding change in class format.
  • 12.3: Due to university closures for the coronavirus, upcoming classes will take place in an alternate format (not on campus). Please check back here for more information soon.
  • 24.2: There is no class today (Monday, 24 Feb.)
  • 18.2: Torgeir Aambø has written up a cleaner version of the solution to exercise #15 from chapter 2.
  • 12.2: Please complete (within the next week) this short survey regarding this course: https://forms.gle/BhYJVmx1CDmtqota8 . Your comments are anonymous, if you choose, and will be used to improve the quality of the course.
  • 3.2: Please email by Feb 6 if you are interested in being a part of the reference group for this semester.
  • 15.1: Just a reminder that there will be no lecture on 16.1 and 20.1; see the schedule below for details.
  • 9.1 (corrected 12.1): Class time/place has been set (see below). First meeting is Monday, 13 January, at 14:15 in SBII room 656. A rough outline for the semester schedule has also been put below; check back regularly for updates.

Teaching/lecturing hours:

  • Time: Monday 14:15 – 16:00 and Thursday 8:15 – 10:00
  • Place: Gløshaugen Sentralbygg 2, room 656

Textbook:

  • Authors: M. F. Atiyah and I. G. Macdonald
  • Title: Introduction to commutative algebra
  • Publisher: Westview Press
  • ISBN: 0-201-40751-5

Lecturer:

  • Email: peder.thompson at ntnu dot no
  • Office: SBII, room 826
  • Office hours: Monday 13:15 – 14:00, and also by appointment (just send me an email to set up a time to meet).

Reference group:

  • Johannes Malkenes, Torgeir Aambø, Elias Klakken Angelsen
  • Meeting #1: February 13
  • Meeting #2: March 13
  • Meeting #3: TBD

Exercises:

  • Exercises will be assigned regularly and posted on the website.
  • We will dedicate about 1 day every other week towards discussing and presenting exercises.
  • I ask that each student presents or leads the discussion for at least one exercise during the course of the semester.

Schedule:

Date Material covered / future plans Exercises assigned
13.01 (Monday) Chapter 1: Rings, ideals, prime and maximal ideals Ch. 1 #1, 2, 4, 5, 8, 9, 10, 15, 16, 17(i-iv), 21, 22
16.01 (Thursday) ———— No class ————
20.01 (Monday) ———— No class ————
23.01 (Thursday) Chapter 1: nilradical, Jacobson radical, operations on ideals
27.01 (Monday) Chapter 1: prime ideals, radical of ideal, extension and contraction of ideals. Discussed ch. 1 ex. #1, 2
30.01 (Thursday) Chapter 2: module homomorphisms, submodules, operations, direct sum/product, f.g. modules Ch. 2 #1, 2, 3, 4, 5, 7, 8, 10, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27
03.02 (Monday) Discuss/present additional chapter 1 exercises
06.02 (Thursday) Chapter 2: exact sequences, some homological algebra and resolutions, tensor products
10.02 (Monday) Chapter 2: restriction/extension of scalars, direct limits, Ext, Tor.
13.02 (Thursday) First half: m-torsion functor and local cohomology, Second half: discuss/present chapter 2 exercises (through #13). Huneke paper for further reading on local cohomology: http://homepages.math.uic.edu/~bshipley/huneke.pdf
17.02 (Monday) Discuss/present exercises from Chapter 2. solution to exercise #15 from chapter 2 written up by Torgeir Aambø
20.02 (Thursday) First half: start chapter 3 on Multiplicatively closed sets, ring of fractions, and examples. Second half: finish direct limit exercises from ch. 2 Ch. 3 #1, 2(first part), 5, 6, 10, 11, 18, 19, + more to come (maybe)
24.02 (Monday) ———— No class ————
27.02 (Thursday) Chapter 3: More localization at prime ideals, localization is exact, local properties, prime ideals under localization
02.03 (Monday) First half: support of a module, Serre subcategories, and classification of specialization closed subsets of Spec(A). Second half: Discuss/present ch. 3 exercises Further reading: Representations of Finite Groups: Local Cohomology and Support by Benson, Iyengar, and Krause
05.03 (Thursday) Chapter 5: integral elements and integral ring extensions Ch. 5 #1, 2, 7, 10, 11, 16, 17
09.03 (Monday) Chapter 5: the going-up theorem, examples
12.03 (Thursday) Chapter 5: preparation for the going-down theorem
16.03 (Monday) We meet virtually in Blackboard Collaborate. We will discuss my notes on Theorem 5.16 (the going-down theorem) and my class notes on Noether normalization and Hilbert's nullstellensatz (v2). In addition, you should complete the chapter 5 exercises. Ch. 5 #1, 2, 7, 10, 11, 16, 17.
19.03 (Thursday) There is no meeting as a class. Instead, please complete these out of class assignments. Read chapter 6 and complete exercise #1. Also continue to work on exercises from chapter 5. As needed, use the discussion forum and/or Blackboard Collaborate to discuss questions/solutions for these exercises. Here I have written up a few brief hints on the chapter 5 exercises if needed. If you still have questions, please post a question to the discussion forum on Blackboard (and, if you'd like, also email me to let me know that you have and I will try to reply on Blackboard so that everyone can see the responses if they want). We will discuss the solutions to exercises from chapters 5 and 6 on Monday. Ch. 6 #1
23.03 (Monday) We meet virtually in Blackboard Collaborate. We will finish discussion of Hilbert's nullstellensatz (weak form), and then discuss solutions to the exercises from chapters 5 and exercises from chapter 6. We will go over these solutions during our virtual class meeting.
26.03 (Thursday) There is no meeting as a class. Instead, please complete these out of class assignments. Read chapter 7 (you can the skip part on primary decomposition at the end) and chapter 8. Complete the exercises from these chapters before next week. Ch. 7 #8, 14, 15, 16, and Ch. 8 #2. Also, Torgeir Aambø wrote up a nice proof of Hilbert's nullstellensatz which uses Noether normalization.
30.03 (Monday) We meet virtually in Blackboard Collaborate (2:15-4pm). Chapter 10: completion of a topological abelian group, Cauchy sequences, inverse limits, and completion as an inverse limit. Click here to download class notes from today (30 March).
02.04 (Thursday) We meet virtually in Blackboard Collaborate (9:15-10am). Chapter 10: exactness properties of inverse limits and exactness properties of completions of topological abelian groups. Click here to download class notes from today (2 April). Also, after you have completed chapter 7 & 8 exercises, independently read my solutions for chapter 7 exercises and my solutions for chapter 8 exercises and email me if you have questions or comments. Individually work on chapter 10 exercises #1, 2. Ch. 10 #1, 2
06.04 (Monday) ———— No class - winter/spring break ————
09.04 (Thursday) ———— No class - winter/spring break ————
13.04 (Monday) ———— No class - winter/spring break ————
16.04 (Thursday) We meet virtually in Blackboard Collaborate (8:15-10am). During the first half we discussed chapter 10 exercises #1 and #2 on the non-exactness properties of both completion and inverse limits. During the second half we continued the lecture over chapter 10, including discussion of filtrations, graded rings and modules, and Lemma 10.8 (the proof of which is to be read on your own). You can download the class notes from today here and download the solutions to chapter 10 exercises #1 and 2 here.
20.04 (Monday) We meet virtually in Zoom (2:15-4pm). Chapter 10: We continue with the Artin-Rees Lemma and its consequences. In particular, we prove that completion is exact on the category of finitely generated modules over a Noetherian ring. We show that completion at a maximal ideal produces a local ring. You can download the class notes from today here. Exercise A: Prove that for a field k, the (x)-adic completion of the polynomial ring k[x] is the power series ring k{[x]}. Exercise B: Prove that if (R,m) is a local Artinian ring, then R is m-adically complete.
23.04 (Thursday) We meet virtually in Zoom (8:15-10am). Chapter 10: Theorem of Krull, associated graded rings, the completion of a Noetherian ring is again Noetherian. You can download the class notes from today here. Sign up for an exam time. Exercise C: Prove that the associated graded ring G(A) is a graded ring, and that the associated graded module G(M) is a graded G(A)-module.
27.04 (Monday) We meet virtually in Zoom (2:15-4pm). Finish up the fact that completion of a Noetherian ring is again Noetherian. Discuss remaining exercises from chapter 10. If time, give a brief advertisement for chapter 11 material. Download notes from today's class here: end of chapter 10 notes, notes for overview of chapter 11 (won't be on exam), and solutions to chapter 10 exercises A, B, C Go through all past exercises and material and compare with classmates; if there are past exercises or material that you would like to discuss in class, please email them to me before Tuesday 28.4. Thursday's class will be devoted to review and discussion of past material as requested.
30.04 (Thursday) We meet virtually in Zoom (8:15-10am). Review material and go over remaining questions on exercises / material. You can download some notes I used for the review here. This is the last day to schedule your oral exam for May.
7.05 (Thursday) Open office hour for questions from 9-10am on Zoom. This is not intended to be a lecture but rather a time that you can come and go with specific questions. Join zoom meeting with Meeting ID 658-2793-8809 (password found on blackboard or email me directly).
11-15 May Oral exams held this week (on Zoom). All exams should be scheduled before Thursday 30.4.
2020-05-05, Peder Thompson