Math 602: Commutative Algebra

WF 8:30am-9:45am in Physics 227

Office Hours: TBD

Homeworks There will be weekly homework, posted here.

Material The material covered in this course will be that of a standard commutative algebra course, but maybe not in a standard order. The material covered will roughly be (in no particular order): ideals and operations on them, varieties, localization, Noether normalization, the Nullstellansatz, Groebner bases and computations, Noetherian and Artinian conditions, radicals, primary decomposition, Hilbert polynomials, dimension theory, modules, completion, integral dependence, going up, going down, Dedekind domains and DVRS, flatness. Also, other topics as time permits.

I'll record the actual topics as we go

1/10/2020 Lecture 1. HW1 out. Due 1/24.
1/15/2020 NO LECTURE. Joint meetings.
1/17/2020 NO LECTURE. Joint meetings.
1/22/2020 Lecture 2: Cayley-Hamilton, Integral extensions, Noether normalization,
1/24/2020 Lecture 3: Definition of dimension, Lying over, Going up, Many forms of the Nullstenllensatz . HW2. Due 1/31/2020.
1/29/2020 NO LECTURE. Utah. This will hopefully be the last missed lecture this term.
1/31/2020 Division algorithm, Dickson's Lemma, Groebner bases, Hilbert's basis theorem for k[x_1, ..., x_n]
2/5/2020 Statement of Buchberger's algorithm, radicals, equivalence between finitely generated, reduced k-algebras with k algebraically closed and algebraic sets.
2/7/2020 Spec, Tensor products of modules and algebras, some basic facts about modules. HW 3 OUT
2/12/2020 Localization I
2/14/2020 Localization II, support, annihilators HW 4 OUT. Due 2/21
2/19/2020 Nakayama's Lemma, Consequences of Nakayama, loose ends on integrality and integral closure.
2/21/2020 CLASS CANCELLED DUE TO TINY AMOUNTS OF SNOW
2/26/2020 Noetherian Conditions / Primary decomposition
2/28/2020 Primary decomposition continued / finished. HW 5 OUT. I also have some incomplex notes / exercise on homological algebra. These will be updated as we go (see the supplements section).
3/4/2020
3/6/2020 Krull's Dimension Theorem
3/18/2020 Coronavirus Disruption
3/20/2020 Coronavirus Disruption
3/25/2020
- Hilbert poly I link here
- Hilbert Poly II link here
- Gradings and Filtrations[24:55]
3/27/2020
- Artin-Reese [13:03]
- Aside on the big picture [3:51]
4/1/2020
- Length [28:47]
- Ancillary results on Length [9:01]
- Numerical Polynomials [12:56]
- Hilbert Functions: Structure [19:37]
4/3/2020
- Local rings and asymptotics [27:47]
- Main Theorems of Dimension Theory[19:35]
4/8/2020
- Systems of Parameters and Independence [11:57]
- Dimension and Transcendence Degree[7:50]
- Tying it together[7:35]
- Regular local rings intro[13:34]
- From here[2:49]
4/10/2020
- Grab bag of dimension theory
4/15/2020
4/17/2020
- DVRs[26:10]
- Characterizations of DVRs[14:41]
- A deferred argument[8:46]
- Argument abstracted once more[3:55]
- Dedekind Domains 1[11:37]
4/22/2020
4/24/2020

Textbook(s) The official textbook is Matsumura's Commutative Ring Theory. However, here are some other recommendations

Introduction to Commutative Algebra by Atiyah and Macdonald. I recommend that you buy a copy of this book. It is not a great reference, but it is an excellent source of exercises, and every mathematician should have a copy.
Commutative Algebra with a View Toward Algebraic Geometry by Eisenbud. I really like this book, but its maybe a bit too big and disorganized for an introduction. Its great to read after you know a thing or two, though.
Notes by Hochster. These are an excellent set of course notes written by a master of commutative algebra. I strongly recommend reading them in addition to Matsumura.
A Term of Commutative Algebra These are an excellent set of lecture notes from 18.705 at MIT.
Notes by Milne
Notes by Gathmann
Zariski-Samuel. The ultimate reference.
Kunz. Introduction To Commutative Algebra and Algebraic Geometry. This is available through Duke Libraries. If I had known about this book when the class started, I would have made it the textbook.
Ideals, Varieties and Algorithms by Cox, Little, O'Shea. An electronic copy is available through Duke libraries. This is an excellent source both for the computational aspects of the course and for
A SINGULAR Introduction to Commutative Algebra by Greuel and Pfister. An electronic copy is available through Duke libraries.
The Macaulay2 Book

Computational Component

I think it is pretty useful to actually be able to compute some things in life. So, there will be a computational component to this course. You will be able to use whatever tools you want for this --- none of the computational exercises will be programming intensive at all. They will just be enough to convince you that you can actually compute things explicitly (this is not something that you can easily see from standard treatments of commutative algebra). My favorite choices of tools are the following

Macaualay 2. This appears to be the choice of many commutative algebra and algebraic geometry researchers. For an introduction see the book above.
Singular. This is appears to be the second choice. It has the advantage that there is an entire graduate level algebra textbook written using it (see above), and that textbook is available to us. Also, the Singular terminal can be accessed from any Duke math department machine.
Sage. This is the most comprehensive computer algebra system, and the choice of number theorists. In general, we will not have a need for all the bells and whistles it offers. Also, its commutative algebra procedures call Singular, anyway. Feel free to use Sage if you're comfortable with it, though!

Supplements

In this section I'll collect supplementary papers by others, and notes by myself.

A wonderful translation of Noether's original paper on primary decomposition.

An incomplete introduction to homological algebra via exercises: homological algebra notes