Math 614: Commutative Algebra

Professor Karen E. Smith
East Hall 3074

Fall 2019
Tuesdays and Thursdays at 10--11:30 am in 2866 East Hall.
Office hours Mondays at 1, Thursdays at 2 and by appointment.

Math 614 is a first introduction to commutative algebra for "beta level" PhD students in math.

The main prerequisite is mathematical maturity of a student who has completed the alpha-courses for the math PhD program at Michigan. In particular, Math 593 (especially modules), Math 594 (especially field extensions, group actions), Math 591 (especially point set topolgy, Math 590 is OK too), and Math 596 (especially familiarity with analytic functions and their power series) will be freely assumed. Please consider taking Math 593 instead if you have not taken this course!

Topics for Math 614:
Noetherian rings; the Hilbert Basis Theorem; the prime spectrum of a ring and the functor Spec; integral extensions (including the Lying over, going up and going down theorems); Noether normalization; Hilbert's nullstellensatz; Krull dimension; various functors on the abelian category of modules over a commutative ring (including localization, completion, and extension and contraction of scalars, the Hom functors); Nakayama's Lemma; associated primes and primary decomposition; Artinian rings and modules; systems of parameters and the Cohen-Macaulay property; normal rings; valuation rings, the Artin Rees lemma; the Frobenius map. I also hope to have time to discuss Groebner Bases and have students play around with Macaulay2 but this remains to be seen.

Mel Hochster's Math 614 notes from 2017 is our main source.
You may also find the following to be of use:
Atiyah-Mac Donald's "Introduction to Commutative Algebra," or
Eisenbud's "Commutative ALgebra with a View towards Algebraic Geometry"
Aluffi's "Algebra: Chapter 0," especially if the category theoretic language is unsettling to you.

IBL Classroom
I intend to experiment with having students working together in class through guided worksheets to learn the material, perhaps every Thursday (we will see how it goes) or even more often (if it is going well). Students will be expected to participate in class, work collegially with other students, and keep a record of all their work.

The Daily Update, a summary of what was discussed in class each day, including assignments and quiz announcements.

Worksheets: Please keep your work organized in a notebook (or ideally, latex document!) to be turned in December.
From September 5 on Noetherian rings and Modules.
From September 10 on The Zariski Topology and Functor Spec.
From September 12 on Algebraic Sets.
From September 17 on The Nullstellensatz.
From September 19 on Localization.
From September 24 on Fibers of Spec maps
From September 26 on Commutative Algebras: algebra finite, module finite, integral
From Oct 1 on Dimension and Integral Extensions
From Oct 3 on The Going Down Theorem
From Oct 8 on Noether Normalization
From Oct 10 on Dimension of K-algebras
From Oct 17 on More Properties of K-algebras
From Oct 22 on Components of Noetherian Schemes
From Oct 24 on Valuation Rings
From Oct 29 on Graded Rings
From Oct 31 on Projective Schemes
From November 5 on Review of Tensor Products
From November 7 on Exactness of Functors
From November 12 on Nakayama's Lemma
From November 14 on Support and Associated Primes
From November 19 on Primary Decomposition (Existence)
From November 21 on Primary Decomposition (Uniqueness)
From November 26 on Krull's Intersection and Principal Ideal Theorems
From December 3 on Normal Rings and the Divisor Class Group
From December 5 on Completion: I-Adic Topology
From December 10 on Completion and Inverse Limits

Problem Sets:
Problem Set 1 due September 19
Problem Set 2 due October 8
Problem Set 3
Problem Set 4
Problem Set 5

Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5, Quiz 6, Quiz 7, Quiz 8,

Course Expectations and Grading scheme:
1. Attendence and participation is required. Students will be expected to have completed reading assignments in advance. (25%)
2. A "lab notebook" where you will write up carefully in your own words the solutions to the Worksheets done in class. This will be submitted on the last day of class. (25%)
3. Weekly short quizzes to make sure you fully absorbed key concepts from the worksheets and homework. (25%)
4. Biweekly problem sets that contain more challenging problems. Please do not use the internet to solve these problems! (25%)

In addition, collegial respect for others and academic integrity are of utmost importance in this course. Working respectfully and inclusively, including listening to others' ideas and points of view, is required at all times. You must cite all sources, including help from others, including classmates or websites. In particular, your "lab notebook" from the class worksheets should include the date of each worksheet and the names of the people you worked with. I am not interested in reading solutions to problems that you have found on the internet. I will consider any work submited without appropriate citation or lifted from online sources as a serious violation of Michigan's acadmic integrity policy. An automatic F will be assigned, regardless of the quality of previously submited work.