Lectures:
TuTh 1:25 p.m. - 3:20 p.m. in Vincent Hall 6
Prerequisites:
Math 8211
Instructor:
Christine Berkesch Zamaere
Office: Vincent Hall 250
Email: cberkesc -at- math.umn.edu (For faster response time, please include "Math8212" in the subject line.)
Office hours:
Tuesday 11:00 a.m. - 12:00 p.m.,
or by appointment (in Vincent Hall 250).
Textbook:
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.
Description:
This is the second semester of an introductory course in commutative and homological algebra. We will dicuss topics including dimension theory, Gröbner bases, free resolutions, and the Cohen--Macaulay property.

Policies for Spring 2015

Assessment:
The course grade will be based on the homework and a presentation.
Homework:
You must collaborate on the homework. For the entire semester, you will work in a small group of 2 to 4 people. From this list of problems, your group should submit 20 problems, which will be collected together on the last day of class, March 26. 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/or agrees with the solutions provided. The homework requirements are subject to changes and additions (but will not be made more difficult).
Presentation:
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.
Disabilities:
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.

Important Dates

January 20   First Math 8212 meeting, Homework group email due by 5 p.m.
February 5   Project paper choice due at 5 p.m. by email
March 12, 24, 26   In-class presentations
March 26   Final 8212 meeting, Homework problems due

Lectures

Below is a list of topics and corresponding reading for the semester, to help you prepare for class. Future lecture topics and dates are subject to change.

Date      Topic
January 20   (10.0) Principal ideal theorem
  (10.1 - only Cor. 10.7) Systems of parameters
January 22   (10.2) Dimension of base and fiber
  (10.3) Regular local rings
  (11.1) DVRs
January 27   (13.1) Noether normalization
January 29   (13.3) Finiteness of integral closure
  (1.9) Quick review: Graded rings, Hilbert function, and Hilbert series
  (15.1, 15.2) Monomial orders
February 3   (15.3) The division algorithm
  More on Gröbner bases
February 5   (15.4) Buchberger's Algorithm (or "How to compute Gröbner bases")
  (15.10) Applications: Ideal membership
February 10   (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
February 12   (15.10) Applications: Hilbert polynomial and computing the dimension of a ring
February 17   Minimal free resolutions in local case
  Minimal free resolutions in graded case
  (17.1, 17.2) Koszul complexes
February 19   (17.3) Building the Koszul complex from parts
February 24   (19.1, 19.2) Projective and global dimension
February 26   (19.3) The Auslander-Buchsbaum formula
  (18.1, 18.2) Cohen--Macaulay rings
March 3   More foundations of the Cohen--Macaulay property
March 5   (18.4) Flatness and depth
  Two proofs that the ideal of generic maximal minors is prime
March 10   Summary
March 12   Presentations: (1:25-1:50) Maggie, (1:55-2:20) Shelley, (2:25-2:50) Theo, (2:55-3:20) Steven
March 24   Presentations: (1:25-1:50) Nicole, (1:55-2:20) Daniel, (2:25-2:50) Craig, (2:55-3:20) Yao-Rui
March 26   Presentations: (1:25-1:50) Kim, (1:55-2:20) Christian, (2:25-2:50) Elise, (2:55-3:20) Yuxiang

Useful links for Macaulay2

Links

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