MW 9:45 a.m. - 11:00 a.m. in Vincent Hall 301
Math 8211
Christine Berkesch
Office: Vincent Hall 254
Email: cberkesc -at- math.umn.edu (For faster response time, please include "Math8212" in the subject line.)
Office hours:
M 11:00 a.m. - 12:00 p.m., W 12:00 p.m. - 1 p.m., or by appointment (in Vincent Hall 254).
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 presentation.
All homework must be typed using LaTex. I am willing to provide a template to get you started.

You may collaborate on the homework. For the entire semester, you will work in a small group of 1 to 4 people. Email your list of group members to me by 5 p.m. on January 22. From this list of problems, your group should submit 20 problems, which will be collected together on the last day of class, May 2, in hard copy; an electronic copy should also be sent to my email address at that time. Clearly number the problem with the numbers below, as well as their placement in Eisenbud (if applicable). In placing your name on an assignment with the others in your group, you are agreeing that each person listed has made a substantial contribution to and agrees with the solutions provided. The assigned problems are available here.
Each student will choose research paper on Commutative Algebra or Homological Algebra. A list of possibilities can be found here, but you may choose another paper with approval. If you have a certain topic in mind and cannot find something on the list of suggestions, I am happy to help you select a paper. At the end of the semester, you will give a presentation in class about the paper you have chosen. Email me by 5 p.m. on February 5 with the information on the paper you have chosen.
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.

January 17   First Math 8212 meeting
January 22   Homework group email due
February 5   Presentation topic choice due
April 23 - May 2   In-class presentations
May 2   Final 8212 meeting: homework due

Below is a list of topics and corresponding reading for the semester.

(11.2) Normal rings and Serre's criterion
(11.3) Invertible modules
(13.1) Noether normalization
(13.3) Finiteness of integral closure
(1.9) Quick review: Graded rings, Hilbert function, and Hilbert series
(15.1, 15.2) Monomial orders
(15.3) The division algorithm
More on Gröbner bases
(15.4) Buchberger's Algorithm (or "How to compute Gröbner bases")
(15.10) Applications: Ideal membership
(15.10) Applications: Elimination
(15.10) Applications: Solving polynomial equations
(15.10) Applications: Implicitization/ring map kernel
(15.10) Applications: Radical membership
M2 examples
(15.8) Gröbner bases and flatness
(15.10) Applications: Hilbert polynomial and computing the dimension of a ring
Minimal free resolutions in local case
Minimal free resolutions in graded case
(17.1, 17.2) Koszul complexes
(17.3) Building the Koszul complex from parts
(19.1, 19.2) Projective and global dimension
(19.3) The Auslander-Buchsbaum formula
(18.1, 18.2) Cohen--Macaulay rings
More foundations of the Cohen--Macaulay property
(18.4) Flatness and depth
Two proofs that the ideal of generic maximal minors is prime


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