Basic information
Welcome!
Symbolic powers are a classical topic that appears ubiquitously throughout commutative algebra. This is a course on symbolic powers and their connections to various topics in commutative algebra and algebraic geometry. You might take this class because you like symbolic powers and want to learn more about them, or just as an excuse to learn more commutative algebra. This will be a commutative algebra II of sorts, covering various classical topics within the field using symbolic powers as a connecting mechanism. We will talk about what symbolic powers are, how to compute them (and the difficulties of doing so), and some of the main open problems in the area, including equality, the containment problem, bounding minimal degrees, and finite generation of symbolic Rees algebras. We will discuss how symbolic powers can be used to prove theorems that are not about symbolic powers, but also how to apply various tools from across commutative algebra to studying symbolic powers. Topics will include: Monomial ideals and some basic combinatorial commutative algebra
Some elementary classical algebraic geometry, including projective varieties
Rees algebras
Characteristic p techniques
Regular sequences
Basics of regular rings and Cohen-Macaulay rings.
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: A basic course in commutative algebra, such as Math 902.
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. There will be problem sets, which you are very welcome to (read: encouraged to!) discuss with me and collaborate on with your classmates; you should however write your own solutions. Your final grade will be based on the problems you turn in, although attendance will also be taken into account.
Office Hours: Just drop by my office whenever, or write me an email if you'd like to set up an appointment. I'm usually around between 8:30 am and 5 pm on Tuesdays, Wednesdays, and Thursdays, and occasionally on Mondays and Fridays.
Instructions on how to install Macaulay2
Macaulay2 online, a way to run Macaulay2 without installing it
Appendix A in the class notes
Here are some Macaulay2 files from our lectures:
Lecture 1: intro to Macaulay2.
Lecture 5: Operations on lists.
Lecture 6: Degrees of generators of a symbolic power.
Lecture 8: Complexes in Macaulay2.
Lecture 9: Free resolutions in Macaulay2.
Lecture 19: Monomial ideals in Macaulay2.
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.
I will give you 5 problem sets, some of which might include some Macaulay2 work, and a separate list of introductory Macaulay2 problems. You can include problems from the introductory M2 problems list with any problem set submission. If you've never used Macaulay2 before, you're welcome to pick all your Macaulay2 problems from the introductory Macaulay2 problems list. To receive an A in the course, you must turn in 20 problems you put a reasonable amount of effort into, and satisfying the following:At least 5 of the problems you turn in involve some Macaulay2 work.
You turn in problems from at least 4 different problem sets.
At least half of the problems you turn in involve some amount of writing (as opposed to only Macaulay2 code).
Here are some random resources:
University of Nebraska — Lincoln
Symbolic powers are a classical topic that appears ubiquitously throughout commutative algebra. This is a course on symbolic powers and their connections to various topics in commutative algebra and algebraic geometry. You might take this class because you like symbolic powers and want to learn more about them, or just as an excuse to learn more commutative algebra. This will be a commutative algebra II of sorts, covering various classical topics within the field using symbolic powers as a connecting mechanism. We will talk about what symbolic powers are, how to compute them (and the difficulties of doing so), and some of the main open problems in the area, including equality, the containment problem, bounding minimal degrees, and finite generation of symbolic Rees algebras. We will discuss how symbolic powers can be used to prove theorems that are not about symbolic powers, but also how to apply various tools from across commutative algebra to studying symbolic powers. Topics will include:
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: A basic course in commutative algebra, such as Math 902.
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. There will be problem sets, which you are very welcome to (read: encouraged to!) discuss with me and collaborate on with your classmates; you should however write your own solutions. Your final grade will be based on the problems you turn in, although attendance will also be taken into account.
Office Hours: Just drop by my office whenever, or write me an email if you'd like to set up an appointment. I'm usually around between 8:30 am and 5 pm on Tuesdays, Wednesdays, and Thursdays, and occasionally on Mondays and Fridays.
Course notes
Here are our class notes. I will be updating these throughout the semester. If you find any typos at all, however small, please let me know.Macaulay2 resources
Here are some Macaulay2 files from our lectures:
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. Your submissions should include two files:I will give you 5 problem sets, some of which might include some Macaulay2 work, and a separate list of introductory Macaulay2 problems. You can include problems from the introductory M2 problems list with any problem set submission. If you've never used Macaulay2 before, you're welcome to pick all your Macaulay2 problems from the introductory Macaulay2 problems list. To receive an A in the course, you must turn in 20 problems you put a reasonable amount of effort into, and satisfying the following:
- Macaulay2 introductory problems (tex file)
- Problem Set 1: due February 4 (tex file)
- Problem Set 2: due February 22 (tex file)
- Problem Set 3: due March 11 (tex file)
- Problem Set 4: due April 15 (tex file)
- Problem Set 5: due May 3 (tex file)
Schedule
- Lecture 1 (Tuesday, January 18): What will this class be about? A short intro to Macaulay2. Minimal primes.
- Lecture 2 (Thursday, January 20): Localization, support, and associated primes.
- Lecture 3 (Tuesday, January 25): Associated primes and primary ideals.
- Lecture 4 (Thursday, January 27): Primary decomposition. The definition of symbolic powers.
- Lecture 5 (Tuesday, February 1): Elementary properties of symbolic powers. Operations on lists.
- Lecture 6 (Thursday, February 3): More elementary properties of symbolic powers. Symbolic powers of homogeneous ideals and their initial degrees. An example in Macaulay2.
- Lecture 7 (Tuesday, February 8): Where are we going? A refresher on dimension and height.
- Lecture 8 (Thursday, February 10): The Koszul complex. Regular sequences. How to write complexes in Macaulay2.
- Lecture 9 (Tuesday, February 15): More on regular sequences. Free resolutions in Macaulay2.
- Lecture 10 (Thursday, February 17): Regular rings.
- Lecture 11 (Twosday, February 22): Depth.
- Lecture 12 (Thursday, February 24): Life is really worth living in a Cohen-Macaulay ring.
- Lecture 13 (Tuesday, March 1): Projective space.
- Lecture 14 (Thursday, March 3): Projective Nullstellensatz. Ideals of points.
- Lecture 15 (Tuesday, March 8): Differential powers and the Zariski—Nagata theorem.
- Lecture 16 (Thursday, March 10): The Zariski—Nagata theorem.
- Lecture 17 (Tuesday, March 22): The Zariski—Nagata theorem for projective varieties and in mixed characteristic. Monomial ideals.
- Lecture 18 (Thursday, March 24): Primary decomposition and symbolic powers of monomial ideals.
- Lecture 19 (Tuesday, March 29): The Stanley-Reisner correspondence. Monomial ideals in Macaulay2.
- Lecture 20 (Thursday, March 31): Edge ideals. Introduction to Rees algebras.
- Lecture 21 (Tuesday, April 5): Rees algebras and the associated graded ring.
- Lecture 22 (Thursday, April 7): The associated primes of powers stabilize.
- Lecture 23 (Tuesday, April 12): Equality of symbolic and ordinary powers.
- Lecture 24 (Thursday, April 14): Equality of symbolic and ordinary powers: the relationship with the associated graded ring and Hochster's theorem.
- Lecture 25 (Tuesday, April 19): Equality of symbolic and ordinary powers: dimension 3 RLRs, some results in higher dimension, monomial ideals.
- Lecture 26 (Thursday, April 21): Equality of symbolic and ordinary powers for monomial ideals: the Packing Problem and Montaño and Núñez Betancourt's sufficient condition.
- Lecture 27 (Tuesday, April 26): A crash course on characteristic p commutative algebra.
- Lecture 28 (Thursday, April 28): F-pure and strongly F-regular rings. Tight closure.
- Lecture 29 (Tuesday, May 3): The containment problem over regular rings. A beautiful proof by Hochster and Huneke. Harbourne's Conjecture.
- Lecture 30 (Thursday, May 5): Other fun open problems.
Here are some random resources:
- Macaulay2, a commutative algebra software
- Macaulay2 online, a way to run Macaulay2 without installing it