Mathematics 250B--Commutative Algebra

Tuesdays and Thursdays, 11:10--12:30, 3 Evans

Arthur Ogus


Algebra, especially commutative algebra, is the main computational tool for many important brances of mathematics. including algebraic geometry, number theory, and representation theory. Math 250B is intended to provide the basic tools of commutative algebra needed to begin research in these areas. I will attempt to emphasize the unifying geometric and functorial point of view introduced by Serre and Grothendieck in the latter half of the twentieth century. The main text will be Serge Lang's Algebra, but I will also refer to other texts for commutative algebra, especially David Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry. Students will be expected to be familiar and comfortable with the basic concepts of algebra, including groups, rings, fields, and polynomials.

The course will be graded in a relatively serious manner. We have a homework grader, who will grade some of the problems each week.  Homework will count as approximately 30% of the grade.  I expect to have a midterm and some sort of final examination as well. We belong to exam group 13, so our exam is scheduled for May 12, from 8:00 am till 11:00 am, in case the exam is an in-class exam. There will be a midterm exam on March 1.

Homework is due Mondays by 3 pm, to be left in a box outside 1095 Evans.  (The office may be a bit hard to find, but it is there.)  In addition to the homework, I expect every enrolled student to email me a question every week, due Friday afternoon.  The question may concern any aspect of the course, and should be sent to  me at, with the words "Math 250B question" as the subject.

For information on when and how to reach me, see my home page.Typically office hours will be MWF from 2:10 till 3:00.


Course Plan


  1. Assignment 1
  2. Assignment 2
  3. Assignment 3
  4. Assignment 4
  5. Assignment 5
  6. Assignment 6
  7. Assignment 7
  8. Assignment 8
  9. Assignment 9
  10. Assignment 10
  11. Assignment 11
  12. Assignment 12




Notes on the Yoneda Theorem

Notes on Colimits and Localization

Notes on Lazard's Theorem

Notes on Enough Injectives

Notes on Associated Primes

Midterm (with solutions)

Notes on Flatness: the local criterion and flatness along the fiber

Notes on Constructibility

Notes on Dimension

Notes on Regular Local Rings

Notes on Depth

Notes on Koszul Complexes

Exam Review

Final Exam (with solutions)