Math 250B - Commutative Algebra

InstructorPaul Vojta
LecturesMWF 1–2, Cory 241
Class Number 20490
Office883 Evans
E-Mail:vojta@math.berkeley.edu
Office HoursMWF 11:30–12:30
excluding non-instructional days.
Please email me to set up a time to meet if you cannot make any of these times.
GSI Office HoursThe GSI for Math 250B is Rahul Dalal. He will hold office hours on MTuF 2–3 (omitting Wednesdays) and Thursdays 10–11 and 1–3, in 1041 Evans.
PrerequisitesMath 250A
Generally speaking, I will expect you to know the material covered in last semester's Math 250A. See the table on that courses's bCourses page.
Required TextEisenbud, Commutative algebra with a view toward algebraic geometry, Springer
Catalog Description Development of the main tools of commutative and homological algebra applicable to algebraic geometry, number theory and combinatorics.
Syllabus The course will likely cover the following chapters or parts of the text:
A2.1–2Tensor products
A1Field theory (separability and regular extensions)
A3.1–8, A3.10Homological things
2Localization
3Associated primes and primary decomposition
4Integral dependence and the Nullstellensatz
5Artin-Rees
6Flatness
8–13Dimension theory
16Modules of differentials

This list is subject to change depending on what was done in 250A.

GradingGrades will be based on homework assignments, and possibly in-class quizzes. There will be no final exam, but the last problem set will be due sometime during the week of final exams.
Homework Weekly or biweekly, assigned in class, generally due on Fridays
CommentsI tend to follow the book fairly closely, but try to give interesting exercises and examples.

Handouts

No.DateTitle Download
1January 22 Proof of associativity of tensor products dvi pdf
2March 30 Proof of the Nullstellensatz (revised 2 April) dvi pdf

Homework Assignments

Homework solutions will be posted on bCourses after each assignment has been graded.

No.DueAssignment Notes
11/31 dvi pdf
22/7 dvi pdf
32/14 dvi pdf
42/21 dvi pdf
52/28 dvi pdf
63/6 dvi pdf
73/13 dvi pdf
83/20 dvi pdf Hint: For the last problem – 250A
94/3 dvi pdf Late change: You are required to use the hint on the first question (Exercise 4.17)

The following policies apply to the homework assignments.

  1. Discussion and the exchange of ideas are essential to doing academic work. For assignments in this course, you are encouraged to consult with your classmates as you work on problem sets. However, after discussions with peers, make sure that you can work through the problem yourself and ensure that any answers you submit for evaluation are the result of your own efforts. In addition, you must cite any books, articles, websites, lectures, etc. that have helped you with your work using appropriate citation practices (other than class lectures and corresponding parts of the textbook). Similarly, you must list the names of students with whom you have collaborated on problem sets. (This paragraph was adapted from http://oue.fas.harvard.edu/icb/icb.do?keyword=k18059&pageid=icb.page498261, accessed on 28 January 2013.)
  2. As an exception to the above, you may not collaborate on problems marked “(nc).”
  3. For problems from the book, you may use without proof any lower-numbered exercises from the same section or any exercises from sections covered earlier. For other exercises, you may use without proof any exercises from earlier sections in the book; for the current section(s), ask. For any exercises, if you weren't able to do an exercise or part of an exercise, you may still assume that it holds for subsequent exercises or parts.
  4. Credit may be reduced if you look up the answer in some other source (as opposed to using the material from elsewhere to aid in your understanding), or if you look ahead in the textbook.

Last updated 2 April 2020