TuTh 2:30 p.m. - 3:45 p.m. in Ford Hall 130
Math 8202
Christine Berkesch Zamaere
Office: Vincent Hall 250
Email: cberkesc -at- math.umn.edu (For faster response time, please include "Math8211" in the subject line.)
Office hours:
Tuesday 1:30 p.m. - 2:20 p.m.,
Thursday 1:30 p.m. - 2:20 p.m.,
or by appointment (in Vincent Hall 250).
Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud, 1995. There is a copy on reserve in the Mathematics library in Vincent Hall.
This is an introductory course in commutative and homological algebra. We will dicuss topics including localization of rings, primary decomposition, completions, and dimension theory.

The course grade will be based on the homework and a project.
You are encouraged to collaborate on the homework. When doing so, you may (and should!) turn in a single assignment for a small group of up to 4 people. In placing your name on an assignment with others, you are agreeing that each person listed has made a substantial contribution to the solutions provided.
You are required to work in a groups of 2-3 for this project. The project has two parts, a paper and a presentation. Additional information can be found here.
Students with disabilities, who will be taking this course and may need disability-related accommodations, are encouraged to make an appointment with me as soon as possible. Also, please contact U of M's Disability Services to register for support.

September 2   First Math 8211 meeting
September 18   Project topic choice due
October 7   Project outline due
October 23   Project progress report due
November 18   Project paper rough draft due
November 25   Project paper peer review due
December 4, 9   In-class project presentations
December 9   Project paper final draft due, Final 8211 meeting

Below is a list of topics and corresponding reading for the semester. No topic next to a date means that we will continue with the previous topic. Note that lectures more than a week in advance are subject to change.

Date      Topic   Textbook references
09-02   Introduction and connections, Modules   1.1, 1.2, 1.3, 1.6, 0.3
09-04   Fractions   2.1
09-09   Noetherian and Artinian rings   2.4
09-11   Hilbert's basis theorem   1.4
09-16   Hom and Tensor   2.2
09-18   Associated primes, Prime avoidance   3.1, 3.2
09-23   Primary Decomposition   3.3
09-25   More primary decomposition   3.6, 3.8
09-30   Nakayama's Lemma and the Cayley--Hamilton Theorem, Normalization   4.1, 4.2
10-02   Primes in an integral extension   4.4
10-07   The Nullstellensatz   4.5
10-09   Graded rings, Hilbert functions   1.5, 1.9
10-14   Associated graded rings   5.1
10-16   The blowup algebra, The Krull Intersection Theorem, Free resolutions   5.2, 5.3, 1.10
10-21   Macaulay2, Flat families   demo.m2, 6.1
10-23   Tor, Flatness   6.2, 6.3
10-28   Direct and inverse limits   A6
11-04   Completions, Cohen Structure Theorem   7.1, 7.2, 7.4
11-06   Maps from power series rings   7.6
11-11   Resolutions   A3.2, A3.3, A3.4
11-13   Homotopies and long exact sequences   A3.5, A3.6, A3.7, A3.8
11-20   Derived functors   A3.9, A3.10, A3.11
11-25   Dimension theory   8.1, 9.0
11-27   No meeting: Thanksgiving break  
12-02   Dimension zero, Presentations   9.1
12-04   Presentations  
12-09   Presentations  

Please include the statement of each problem before its solution and use a separate page for each problem.

Assignment     Due date
Problem Set 1   Thursday, September 18
Problem Set 2   Thursday, October 2
Problem Set 3   Thursday, October 16
Problem Set 4   Thursday, October 30
Problem Set 5   Thursday, November 13
Problem Set 6   Thursday, December 4


Christine Berkesch Zamaere  ***  School of Mathematics  ***  University of Minnesota