Commutative Algebra

Home page of A.J. de Jong.

The topics we will discuss are: Spectrum of a ring, elementary properties, flat and integral ring extensions, going up and going down, constructable sets and Chevalley's Theorem, graded modules, Artin-Rees theorem, dimension theory of Noetherian local rings, dimension theory of finitely generated k-algebras and transcendence degree, Hilbert Nullstellensatz. We will attempt to motivate the theory by giving examples from algebraic geometry, but the theorems discussed in the lectures will be theorems of commutative algebra.

I will be using the book by Matsumura, Commutative Algebra (Mathematics Lecture Notes Series ; 56), Benjamin-Cummings Pub Co; 2d ed edition (July 1980). It is the book I learned this material from. This book is out of print. On the other hand, you can find all the material covered in any reasonable commutative algebra books. It is a good idea to choose one and stick to it however. Come and find me in my office if you are not sure which books to look at; I have a bunch of them available for you to look at. Another possibility is to use the lecture notes by Professor Robert Friedman from his previous years teaching this course. They are available to registered students through courseworks.

It is strongly encouraged to go to the lectures, which are on Tuesday and Thursday 2:40-3:55 in Mathematics 520.

Problem sets will be announced in lecture on Tuesdays and on this web page. They are due in lecture on the next Tuesday. Please write out all arguments completely. Late policy: you can hand in on Wednesday and Thursday as well, but each day late costs you 20% of your score. Please hand in late homeworks in the TA's mailbox.

The TA for the course is Matt Deland. His email address is deland@math.columbia.edu and his office is Mathematics 408. He will have office hours on Monday probably between 10-11AM. Details TBA.

Grades are computed by a weighted average between the scores on problem sets and final. The weights are 2/3 and 1/3 respectively.

The final will be a written exam.

Here are the weekly problem sets. Please hit the refresh button on your browser to make sure you have to latest list.

• First problem set due on Tuesday September 12: Problems -2,-1,0,1,2,3,4,5,6,7,8,9,10 from set-1 below.
• Second problem set due on Tuesday September 19: Problems 11-19 from set-1 below.
• Third problem set due on Tuesday September 26: Problems 1-6 from set-2 below.
• Fourth problem set due on Tuesday October 3: Problems 1-5 from set-3 below.
• Fifth problem set due on Tuesday October 10: Problems 1-3 from set-4 below. (Changed on Sunday, October 8, 2006.)
• Sixth problem set due on Tuesday October 17: Problems 1-3 from set-5 below.
• Seventh problem set due on Tuesday October 24: Problem 4 of set-6 below and your choice of one of the problems 2 or 3 from set-6.
• Eigth problem set due on Tuesday October 31: Problems 1-7 of set-7 below.
• Ninth problem set due on Tuesday November 14 (note that we are skipping a week but that this is a fairly long set): Problems 1,2,3, and 5(b) as well as one of 4 or 6 of set-8 below.
• Tenth problem set due on Tuesday November 21: Any subset of 4 problems of set-9 below, except for the set {1,2,3,4}.
• Eleventh problem set due Tuesday November 28: Choose 3 among the 7 problems of the collection set-10 below. (Note: Problem 7 is a kind of research problem.)
• Twelth problem set due Tuesday December 6: Problems 1,2,3 of set-11 below.

Downloads:

• set-1.pdf set-1.dvi set-1.tex set-1.ps
• set-2.pdf set-2.dvi set-2.tex set-2.ps
• set-3.pdf set-3.dvi set-3.tex set-3.ps
• set-4.pdf set-4.dvi set-4.tex set-4.ps
• set-5.pdf set-5.dvi set-5.tex set-5.ps
• set-6.pdf set-6.dvi set-6.tex set-6.ps
• set-7.pdf set-7.dvi set-7.tex set-7.ps
• set-8.pdf set-8.dvi set-8.tex set-8.ps
• set-9.pdf set-9.dvi set-9.tex set-9.ps
• set-10.pdf set-10.dvi set-10.tex set-10.ps
• set-11.pdf set-11.dvi set-11.tex set-11.ps