Commutative Algebra I | UNL Fall 2022

Commutative Algebra I

Fall 2022

University of Nebraska — Lincoln

Instructor: Eloísa Grifo (please address me as Eloísa)

Class notes   If you find any typos at all, however small, please let me know.


Basic course info: This is a first course in commutative algebra covering core topics in the field including noetherian rings, graded rings, localization, Nakayama's lemma, integral extensions, primary decomposition, Hilbert functions, and dimension theory.

Computing examples can be quite difficult. We will learn a little bit about how we can use Macaulay2 to do calculations; I will not assume you've ever used Macaulay2 before. As surprising as it may sound, we can use a computer to do very abstract calculations!

Prerequisites: Basic ring and module theory, such as the material in Math 817/818.

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 and the Macaulay2 documentation. The midterm and final exam will be take-home and look a lot like the problem sets, except you cannot collaborate on the problems nor use the class notes.

Office Hours: Just drop by my office whenever, or write me an email if you'd like to set up an appointment. You are most likely to find me in my office on days we have class.
I have office hours for my undergraduate class on Mondays 4-5 and Wednesday 2-3. Those are times you are certain to find me in my office, though any undergraduate students who come by at that time will have to take priority.


Macaulay2 resources

  • Instructions on how to install Macaulay2
  • Macaulay2 online, a way to run Macaulay2 without installing it

    • Here are some Macaulay2 files from our lectures:

  • Lecture 1: intro to Macaulay2.
  • Lecture 2: Modules.
  • Lecture 3: Algebras.
  • Lecture 12: Graded rings.
  • Lecture 43: Gröbner basis.


    • Problem Sets

    Instructions: You are welcome to work together 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 each problem set on canvas; each submission will include two files:
  • The written problems should be in 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.
  • Any Macaulay2 work you want to submit should preferably be in a .m2 file. If you are a windows user and have to use Macaulay2 online, please copy your work (inputs only) to a txt file, and then rename it to be a .m2 file. You can include written comments on your Macaulay2 work in your pdf as well.

    • Schedule

    • Lecture 1 (Monday, August 22): Modules. A short intro to Macaulay2.
    • Lecture 2 (Wednesday, August 24): Modules and how to set them up in Macaulay2. Algebras.
    • Lecture 3 (Friday, August 26): Algebras and how to set them up in Macaulay2.
    • Lecture 4 (Monday, August 29): Module-finite vs algebra-finite.
    • Lecture 5 (Wednesday, August 31): Integral closure.
    • Lecture 6 (Friday, September 2): The relationship between module-finite, algebra-finite, and integral.
    • Lecture 7 (Wednesday, September 7): Module-finite is equivalent to algebra-finite and integral.
    • Lecture 8 (Thursday, September 8): A very short introduction to homological algebra. Noetherian rings.
    • Lecture 9 (Friday, September 9): Noetherian modules.
    • Lecture 10 (Monday, September 19): Hilbert's Basis Theorem.
    • Lecture 11 (Wednesday, September 21): The Artin-Tate Lemma. An application to a classical problem about invariant rings.
    • Lecture 12 (Friday, September 23): Finishing up the proof that invariant rings of finite groups are finitely generated algebras. Graded rings.
    • Lecture 13 (Monday, September 26): Graded rings and how to set them up in Macaulay2.
    • Lecture 14 (Wednesday, September 28): Graded maps. Graded rings and a condition for algebra-finiteness.
    • Lecture 15 (Friday, September 30): Noetherian graded rings.
    • Lecture 16 (Monday, October 3): Another application to invariant rings.
    • Lecture 17 (Wednesday, October 5): Prime and maximal ideals.
    • Lecture 18 (Friday, October 7): Spec and radical ideals.
    • Lecture 19 (Monday, October 10): More about radicals.
    • Lecture 20 (Wednesday, October 12): Affine varieties.
    • Lecture 21 (Friday, October 14): The coordinate ring of a variety.
    • Lecture 22 (Wednesday, October 19): Nullstellensatz.
    • Lecture 23 (Thursday, October 20): Local rings and localization.
    • Lecture 24 (Friday, October 21): NAK.
    • Lecture 25 (Monday, October 24): NAK and minimal generators. Finiteness of minimal primes.
    • Lecture 26 (Wednesday, October 26): Support of a module.
    • Lecture 27 (Friday, October 28): Associated primes.
    • Lecture 28 (Monday, October 31): Existence of associated primes.
    • Lecture 29 (Wednesday, November 2): Finiteness of associated primes.
    • Lecture 30 (Friday, November 4): Prime avoidance. Primary ideals.
    • Lecture 31 (Monday, November 7): Many characterizations of primary ideals.
    • Lecture 32 (Wednesday, November 9): Primary decomposition: existence.
    • Lecture 33 (Friday, November 11): Primary decomposition: uniqueness theorems.
    • Lecture 34 (Monday, November 14): Dimension and height.
    • Lecture 35 (Wednesday, November 16): Dimension and height.
    • Lecture 36 (Friday, November 18): More about dimension.
    • Lecture 37 (Monday, November 21): Lying Over, Incomparability, and Going up.
    • Lecture 38 (Monday, November 28): Going Down.
    • Lecture 39 (Wednesday, November 30): Noether normalizations.
    • Lecture 40 (Friday, December 2): Nice properties of finitely generated k-algebras. Preparing for Krull's Height theorem.
    • Lecture 41 (Monday, December 5): Krull's Height theorem.
    • Lecture 42 (Wednesday, December 7): Systems of parameters.
    • Lecture 43 (Friday, December 9): Gröbner basis.


    Here are some random resources:

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