Basic information
Welcome to Commutative Algebra II!
This course is a sequel to Math 905 (Commutative Algebra I, last offered in Fall 2024). The course should be accessible to students who have learned some basic commutative algebra such as in the content of my Math 905 notes (e.g., basics of noetherian rings, localization, primary decomposition, and Krull dimension). Also, a course in homological algebra (such as Math 915) is essential; we will reference my Math 915 notes often. In particular, we will assume knowledge of basic homological algebra and some familiarity with Ext and Tor.
Along the way we will do lots of computations: of free resolutions, of depth and grade, of dimension, and so on. We will use Macaulay2 to do some of these calculations.
Course expectations: Attendance is expected, as much as possible, but please do NOT attend class if you are feeling ill or may have been exposed to COVID-19. You are encouraged to work on the problem sets together in groups, or discuss them with me; you should however write up your own solutions. The only other resources you are allowed to use to solve the problem sets are our class notes. The final exam will be take-home and look a lot like the problem sets, except you cannot collaborate on the problems, and you are encouraged to try to solve the problems without checking the class notes.
Office Hours: Mondays 5 — 6 pm and Fridays 9 — 10 am.
Apart from these notes, we will not follow any particular textbook, but recommended sources include:
Bruns and Herzog's Cohen-Macaulay rings
Eisenbud's Commutative Algebra with a view towards algebraic geometry
I will be posting problems as we go, and each problem set will remain incomplete until about one week before the deadline.
Final Problem Set 1: due Wednesday, January 27 (tex)
Incomplete Problem Set 2: due Wednesday, February 18 (tex)
Lecture 1 (Monday, January 12): Free resolutions.
Lecture 2 (Wednesday, January 14): More on free resolutions. Graded free resolutions.
Lecture 3 (Friday, January 16): A quick introduction to complexes with Macaulay2. Worksheet 1: free resolutions.
Lecture 4 (Wednesday, January 21): The Koszul complex.
Lecture 5 (Friday, January 23): Properties of Koszul homology.
Lecture 6 (Monday, January 26): A took for inductive proofs with the koszul complex. Intro to regular sequences.
Lecture 7 (Wednesday, January 28): More about regular sequences.
Lecture 8 (Friday, January 30): Regular local rings.
Lecture 9 (Monday, February 2): Auslandar--Buchsbaum--Serre.
Lecture 10 (Wednesday, February 4): Global dimension.
Lecture 11 (Friday, February 6):
Lecture 12 (also Friday, February 6):
Worksheet 0: a hands on refresher on heights and dimension
Worksheet 1: Free Resolutions
An introduction to complexes and free resolutions in Macaulay2
Macaulay2, a commutative algebra software
Macaulay2 online, a way to run Macaulay2 without installing it
nLab, a math wiki written from the perspective of (higher) category theory
University of Nebraska-Lincoln
This course is a sequel to Math 905 (Commutative Algebra I, last offered in Fall 2024). The course should be accessible to students who have learned some basic commutative algebra such as in the content of my Math 905 notes (e.g., basics of noetherian rings, localization, primary decomposition, and Krull dimension). Also, a course in homological algebra (such as Math 915) is essential; we will reference my Math 915 notes often. In particular, we will assume knowledge of basic homological algebra and some familiarity with Ext and Tor.
Along the way we will do lots of computations: of free resolutions, of depth and grade, of dimension, and so on. We will use Macaulay2 to do some of these calculations.
Course expectations: Attendance is expected, as much as possible, but please do NOT attend class if you are feeling ill or may have been exposed to COVID-19. You are encouraged to work on the problem sets together in groups, or discuss them with me; you should however write up your own solutions. The only other resources you are allowed to use to solve the problem sets are our class notes. The final exam will be take-home and look a lot like the problem sets, except you cannot collaborate on the problems, and you are encouraged to try to solve the problems without checking the class notes.
Office Hours: Mondays 5 — 6 pm and Fridays 9 — 10 am.
Course notes
Here are the Course notes. I will be updating these throughout the semester. If you find any typos at all, however small, please let me know.Apart from these notes, we will not follow any particular textbook, but recommended sources include:
Problem Sets
Instructions: You are welcome to work together with your classmates on the problems, and I will be happy to give you hints or discuss the problems with you, but you should write up your solutions by yourself. You will submit your work on canvas; your submission should be a pdf file, but you are not required to type your solutions. If you prefer to handwrite your work on paper, there are apps like Scannable or Genius Scan that can scan your work into one pdf.I will be posting problems as we go, and each problem set will remain incomplete until about one week before the deadline.