Basic information
Welcome to Commutative Algebra!
This course will be an introduction to commutative algebra, and will continue through the Spring quarter with a course called Homological Algebra.
This first course is an introduction to commutative algebra, a subject that has interactions with algebraic geometry, number theory, combinatorics, category theory, representation theory, and several complex variables. The emphasis will be on Noetherian rings. Topics will include the Noetherian property, integral extensions, Hilbert's Nullstellensatz and the basic dictionary between commutative algebra and algebraic geometry, Noether normalization, localization, chains of prime ideals, Krull dimension, Artinian rings, normal Noetherian rings, primary decomposition, and the Krull height theorem. We will learn and use some elementary homological algebra throughout the course.
The course will also include a brief introduction to Macaulay2. Macaulay2 is a software system devoted to supporting research in algebraic geometry and commutative algebra. We will briefly discuss how one can use computer software in pure math research, and try some very elementary examples in Macaulay2.
Prerequisites: The official prerequisite is the Algebra sequence at UCR. If you have taken an introduction to ring theory somewhere else, I will allow you to enroll as long as you convince me you have enough background to follow. If you are less than comfortable with the definition of a module over a ring (or lack some guiding examples) and/or are less than somewhat OK with free modules, I encourage you to quickly review these notions. While we will introduce them in class, having some familiarity with these notions will be helpful. A friendly source is Dummit and Foote, 10.1-10.4. It will also be helpful, but not required, that you have some familiarity with exact sequences.
Course expectations: Attendance is expected. There will be 5 problem sets, which you are very welcome to (read: encouraged to!) work on together in groups; you should however write your own solutions to the problem sets. Your final grade will be based on the problem sets, although attendance will also be taken into account.
Office Hours: Fridays 11 am to 1 pm and by appointment.
Introduction to Commutative Algebra, by Atiyah and Macdonald
Mel Hochster's lecture notes (the Fall 2020 version)
Jack Jeffries's lecture notes (2018 and 2019 versions), and
Commutative Ring Theory, by Matsumura
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, a print screen of your work is fine. You can include any written comments on your Macaulay2 work in your pdf as well.
Here are some random commutative algebra resources:
University of California, Riverside
This course will be an introduction to commutative algebra, and will continue through the Spring quarter with a course called Homological Algebra.
This first course is an introduction to commutative algebra, a subject that has interactions with algebraic geometry, number theory, combinatorics, category theory, representation theory, and several complex variables. The emphasis will be on Noetherian rings. Topics will include the Noetherian property, integral extensions, Hilbert's Nullstellensatz and the basic dictionary between commutative algebra and algebraic geometry, Noether normalization, localization, chains of prime ideals, Krull dimension, Artinian rings, normal Noetherian rings, primary decomposition, and the Krull height theorem. We will learn and use some elementary homological algebra throughout the course.
The course will also include a brief introduction to Macaulay2. Macaulay2 is a software system devoted to supporting research in algebraic geometry and commutative algebra. We will briefly discuss how one can use computer software in pure math research, and try some very elementary examples in Macaulay2.
Prerequisites: The official prerequisite is the Algebra sequence at UCR. If you have taken an introduction to ring theory somewhere else, I will allow you to enroll as long as you convince me you have enough background to follow. If you are less than comfortable with the definition of a module over a ring (or lack some guiding examples) and/or are less than somewhat OK with free modules, I encourage you to quickly review these notions. While we will introduce them in class, having some familiarity with these notions will be helpful. A friendly source is Dummit and Foote, 10.1-10.4. It will also be helpful, but not required, that you have some familiarity with exact sequences.
Course expectations: Attendance is expected. There will be 5 problem sets, which you are very welcome to (read: encouraged to!) work on together in groups; you should however write your own solutions to the problem sets. Your final grade will be based on the problem sets, although attendance will also be taken into account.
Office Hours: Fridays 11 am to 1 pm and by appointment.
Course notes
Here are the course notes. I will be updating these throughout the quarter. 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: For each problem set, pick 5 problems to submit, including at least one problem that involves some Macaulay2 work. 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. Your submission should have two parts:- Problem Set 1: due January 20
- Problem Set 2: due February 3 (tex)
- Problem Set 3: due February 17 (tex)
- Problem Set 4: due March 3 (tex)
- Problem Set 5: due March 17 (tex)
Schedule
- Monday, January 4: Background. Noetherian rings. M2 examples
- Wednesday, January 6: Noetherian rings and modules.
- Monday, January 11: Hilbert's basis Theorem. Algebra-finite and module-finite extensions. M2 examples
- Wednesday, January 13: Integral extensions and characterization of module-finite extensions. An application to rings of invariants.
- Wednesday, January 20: Graded rings and graded modules. M2 examples
- Monday, January 25: Noetherian graded k-algebras and an application to invariant rings. Systems of equations and varieties.
- Wednesday, January 25: Recap of basic facts about prime and maximal ideals. Hilbert's Nullstellensatz. M2 examples
- Monday, February 1: The Strong Nullstellensatz. The Algebra-Geometry dictionary.
- Wednesday, February 3: The Zariski Topology and the spectrum of a ring.
- Monday, February 8: Local rings. Localization.
- Wednesday, February 10: Localization of a module. NAK.
- Wednesday, February 17: Graded NAK and minimal generators. Minimal primes and support. M2 examples
- Monday, February 22: Associated primes. Prime Avoidance.
- Wednesday, February 24: Primary Decomposition.
- Monday, March 1: 100 years of Noether's Idealtheorie in Ringbereichen: existence and uniqueness of primary decompositions in Noetherian rings.
- Wednesday, March 3: Height and dimension. Artinian rings.
- Monday, March 8: Krull's Height Theorem.
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Wednesday, March 10: Over, up, and down. Noether normalizations. Hilbert functions.
Here are some cool notes by Alexandra Seceleanu about Hilbert functions.
Here are some random commutative algebra resources:
- Why every ring should have a 1
- Macaulay2, a commutative algebra software
- Macaulay2 online, a way to run Macaulay2 without installing it
- commalg, a website with conference listing and updates from the commutative algebra community