Math 538, Commutative Algebra - Spring 2021

Instructor: Jack Huizenga

Jack Huizenga
Office: 324 McAllister
  • Meeting times: MWF 11:15-12:05 on Zoom. The link is on Canvas.
  • First meeting: Wednesday, January 20
  • Office hours: After class each day, or by appointment.
  • Textbooks: Aityah and MacDonald, "Introduction to Commutative Algebra" will be the main text for the course. We will also cover some material from Eisenbud, "Commutative Algebra with a View Toward Algebraic Geometry."
Brief course description: We will cover the majority of the Atiyah-MacDonald textbook. The textbook is a terse introduction to commutative algebra, so some extra material included in the exercises or from the Eisenbud textbook will also be presented. This is a first course in commutative algebra, which assumes some basic concepts from ring theory are already known. Topics covered include commutative rings, ideals, modules, localization, primary decomposition, integral extensions, Noetherian rings, the Nullstellensatz, Artinian rings, DVRs and Dedekind domains, completions, and dimension theory. A secondary goal of the course is an introduction to the dictionary between commutative algebra and algebraic geometry; many of these connections will be further explored in exercises.
Prerequisites: Math 536, Abstract Algebra. While at the surface most of our treatment will be self-contained, previous encounters with rings and modules are likely essential to keep up in the course. The "middle chapters" of Dummit and Foote "Abstract Algebra" are a good reference for any lacking background.
Course Delivery: The course will be presented online via Zoom in a remote synchronous mode. Students are strongly encouraged to both ask and answer questions, and due to the relatively small size of the class it is encouraged to share cameras. The instructor will record lecture notes on a tablet device and post the notes from class each day. Recordings of lectures will also be posted after class, but in-class attendance is strongly encouraged whenever feasible. You and other students will and *should* have many questions to be answered, and you should always feel free to interrupt when something is not clear. If you have a question, many of your fellow students probably have the same question, and we can save a great deal of time and confusion by addressing it.
Homework: There will be approximately 7 problem sets, due at the beginning of class every other Wednesday. Homework sets will be long, since it is intended that a student who completes all the exercises will be ready to use the material in further courses and research. I strongly encourage all students to complete the homework, especially if commutative algebra is likely to be relevant to your eventual research specialty. This material is challenging, and it is not possible to learn the subject "by osmosis." However, I understand that you have many competing demands on your time (especially as you progress in your Ph.D. studies). You are all adults, so you may determine your own goals for the course and complete the homework accordingly.
  • To achieve working proficiency in commutative algebra you should complete all the homework on time, in order to keep up with the class.
  • To become acquainted with some of the basic ideas of commutative algebra, you should at a minimum work on a seletion of problems from each topic and submit a collection of roughly 50 exercises by the end of the semester. (If you fall into this group it may be particularly helpful to work with others.)
  • If you already have substantial prior experience with commutative algebra, it might be most appropriate to look at the exercises and verify for yourself that you know the key insights.
If you already have a Ph.D. advisor, it would be a good idea to consult with them and discuss how much time you should spend on this course. Late work will be accepted, but will be marked for completion only. Students are encouraged to work on the problems together as desired, but your final writeup should be completed independently.
Grading and Expectations: Your grade will be based on a combination of your homework and attendance.
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