## Math 508: Advanced commutative algebra

#### Winter quarter 2016-7

Lectures: MWF 9:30-10:20 in PDL C-401
Instructor: Jarod Alper (jarod@uw.edu)
Office: PDL C-328

Syllabus
Course summary

Textbooks: The main textbook for the class will be:
• Cohen-Macaulay rings, Winfried Bruns and Jürgen Herzog.
I will be assuming a certain familiarity with the concepts from Chapters 1-9 of
• Introduction to Commutative Algebra, M.F. Atiyah and I.G. Macdonald.
I highly recommend this book if you want to review basic commutative algebra. Additionally, we will cover some of the material of Chapters 10-11 in detail. Other references that may be useful are:
• Commutative Algebra with a View Toward Algebraic Geometry, David Eisenbud
• Commutative Ring Theory, Hideyuki Matsumura

Macaulay2: In this class, we will experiment with the use of the Macaulay2 software system. Macaulay2 is a mathematical sotware system which is particularly designed for computing objects in commutative algebra and algebraic geometry. While the main goal of this course is to discuss important deep theorems in commutative algebra and provide rigorous proofs, my hope is that we will be able to further our intuition of these concepts through computations in Macaulay2.

In some of the homework exercises, you will be asked to compute examples using Macaulay2. Even if you have no experience whatsoever in programming, you should have no problem in quickly learning how to use Macualay2 as the language is extraordinarily intuitive for a mathematician.

During the second week of class, I will run a tutorial on how to compute some basic examples using Macaulay2. Before the tutorial, I highly recommend you install Macaulay2 and make sure it works by computing something like 1+1. If you have a laptop, it may be useful to bring it to the tutorial so that you can also play around. Also, I have found it useful to use the emacs text editor as an interface for Macaulay2. If you want to do this, read here.

I should emphasize that I am not an expert in Macaulay2. Recently I have found it extremely useful in my research to compute examples and gain intuition (but not as a means to prove theorems). Hopefully, as a class, we can explore together the capabilities of Macaulay2.

Update: Here is the Macaulay2 code from the tutorial.

Assignments: