MMA330 Commutative Algebra Autumn 24

 

Welcome to MMA330 Commutative Algebra!

 

The first retake exam, together with sketched solutions, can be found here.

 

The exam, together with sketched solutions, can be found here.

 

If you are taking this course, please read the information below carefully.

The lecturer/examiner for the course is Christian Johansson. If you have questions regarding the course, my email address is chrjohv@chalmers.se.

The schedule of the course is as follows:

• Mondays 13.15-15.00 in MVH12
• Tuesdays 13.15-15.00 in Pascal
• Thursdays 15.15-17.00 in MVH12 (except Sep 26 and Oct 17 when the time is 13.15-15.00)

The schedule is now on TimeEdit.

The course book is “Undergraduate Commutative Algebra” by Miles Reid. I expect that we will cover roughly chapters 1-4, 6 and 8, with a few omissions and additions. We will have three sessions per week, and I expect to spend roughly two thirds of the time explaining the theory and the remaining third going over exercises (but we will mix theory and exercises during the sessions).

The course prerequisite is some basic ring theory, as in MMG500 "Algebraic Structures", but this not logically necessary to follow the course. In practice, I will recall all the definitions, but I will expect that you have seen the definitions of

• Rings, ideals (including prime and maximal ideals) and quotient rings,
• Integral domains and their field of fractions,
• Principal ideal domains, Euclidean domains, and Unique Factorization Domains,
• Fields

and have some familiarity with these notions. Before the first lecture, I strongly recommend reading Chapter 0 in Reid's book, and to read up on the notions above. The course will start with Chapter 1.

 

The following are the recommended exercises for the course, from Reid's book, ordered by chapter. There might be additional exercises added in later.

• 0.3, 0.7, 0.8, 0.9 
• 1.1-1.17 
• 2.1, 2.2, 2.6-2.10, 2.14 
• 3.1-3.6, 3.8 
• 4.1-4.6 
• 6.1-6.5, 6.11-6.13 
• 8.1-8.4

I expect that the exam questions will be broadly similar to the above problems. As mentioned above, I recommend everyone to read Chapter 0 on your own to better understand the aims of Reid's book, and to gain a perspective on commutative algebra shared by many (but not all!) mathematicians. I will try to add other perspectives as we go along. As we progress during the course, I also recommend you to start looking at Chapter 9 as well to see glimpses of where commutative algebra goes beyond a first course. 

 

Program

The schedule of the course will be in TimeEdit. I will fill out the following schedule detailing what we have done as we progress in the course.

Schedule:

Day	Sections	Content
Mon 2/9		Introduction, historical motivation for ideals
Tue 3/9	1.1-1.5	Ideals, prime and maximal ideals, the spectrum as a set
Thu 5/9	1.6-1.10, 1.12-1.15	Existence of prime ideals, radicals, local rings
Mon 9/9	2.1, 2.2	Exercises from Ch 1, definition of modules
Tue 10/9	2.3-2.7	Isomorphism Theorems, the Cayley-Hamilton Theorem
Thu 12/9	2,9,2.10, 3.1, 3.2, 3.4	Exact sequences, Noetherian rings and modules
Mon 16/9	3.2-3.4	Exercises from Ch 2, more on Noetherian modules
Tue 17/9	3.5, 3.6	Exercises from Ch 2, Hilbert Basis Theorem
Thu 19/9	6.1	Invariant Theory, definition of localization
Mon 23/9	6.2-6.3	Exercises from Ch 3, ideals in localizations
Tue 24/9	6.4	Exercises from Ch 3, ideals in localizations (continued)
Thu 26/9	6.5-6.7, 4.1-4.2	Localization of modules, finiteness and integrality in ring extensions
Mon 30/9	4.3	Exercises from Ch 6, finiteness/integrality
Tue 1/10	4.4, 4.6, 4.9	Exercises from Ch 6, integral closure, Noether normalization
Thu 3/10	Eisenbud (pdf)	General Nullstellensatz Eisenbud
Mon 7/10	5.2-5.3,5.5-5.6	Exercises from Ch 4, Nullstellensatz
Tue 8/10	Stacks project link	Exercises from Ch 4, Local Artinian rings
Thu 10/10	8.1-8.4	Discrete valuation rings
Mon 14/10	8.7-8.8	More on normality
Tue 15/10	Cohn	Exercises from Ch 8, modules over a PID (non-examinable)
Thu 17/10		Modules over a PID, Jordan normal form
Mon 21/10		Tentor 201026, 210104
Tue 22/10		Övningar kap 1, tenta 210815
Thu 24/10		Overflow

 

Back to the top

Other sources:

There are many excellent textbooks on commutative algebra. Below is a sample of longer and/or more advanced texts that can be used as a complement if you wish to see other treatments of the topics from the course (or you are curious about more advanced material).

• Atiyah and MacDonald. Introduction to commutative algebra.
• Cohn. Classic Algebra, Basic Algebra and Further Algebra and Applications.
• Eisenbud. Commutative algebra with a view towards algebraic geometry.
• Matsumura. Commutative Ring Theory.

Finally, there is the Stacks Project, which is an online encyclopedia/reference work on commutative algebra and algebraic geometry (but it is *not* a textbook!). 

Old exams

January 2023, Solutions

October 2022, Solutions

2020-2021, Solutions 1, Solutions 2

2016-2019, Solutions

 

 

Back to the top