Commutative Algebra Autumn 2021

Prof. Emmanuel Kowalski
Ilaria Viglino
Wed 16-18, HG E 1.1
Fri 8-10, HG E 5
Exercise classes
Group 1: Thu 9-10, HG E 1.2
Group 2: Thu 12-13, HG E 1.2

Starting dates
First lecture: Wed, September 22, 2021
First exercise class: Thu, September 23, 2021

The last lecture is on December 22

Format of the lectures

The lecture will be in presence, at the blackboard. The multimedia equipment of the lecture rooms will only allow the sound to be livestreamed and recorded. For students unable to attend the lecture, the notes will be available as PDF, usually in advance of the lecture. Moreover, the basic reference book is also freely available for study.
Livestream of the Wednesday lecture (HG E 1.1)
Livestream of the Friday Lecture (HG E 5)
Audio recording of the lectures

Format of the exam

The exam will be held online (with Zoom). The link for this will be communicated later.
The rules of the exam ares explained on the Forum, as well as in this file.


This course provides an introduction to the fundamental ideas of commutative algebra, in particular (but not exclusively) as a foundation towards algebraic geometry.

Topics include:


Algebra I and II (or a similar introduction to the basic concepts of ring theory, including Principal Ideal Domains and factorial rings, as well as elementary Galois theory).


A DMATH forum is available for discussion and to ask/answer questions about the course, see here.


The first exercise class on September 23 will cover some reminders of basic facts and definitions.

The first exercise sheet will be posted here on Friday afternoon, September 25. The following exercise sheets will usually be posted on Fridays, and sometimes on the following Monday morning. We expect you to look at the problems and to prepare questions for the exercise class on the following Thursday.

Please hand in your solutions by the following Friday. You should submit your exercise sheets in PDF form to the assistant in charge of your group. Written solutions will be provided the day after the deadline.

Exercise sheet due by Solutions
Exercise sheet 1 October 1 Solutions 1
Exercise sheet 2 October 15 Solutions 2
Exercise sheet 3 October 29 Solutions 3
Exercise sheet 4 November 12 Solutions 4
Exercise sheet 5 November 26 Solutions 5
Exercise sheet 6 December 10 Solutions 6
Exercise sheet 7 December 24 Solutions 7

Exercise classes

Th 09-10HG E 1.2Younghan BaeEnglish
Th 12-13HG E 1.2Ilaria ViglinoEnglish

Content of the Lectures

We will summarize below the contents of each lecture, with references to the book of A. Chambert-Loir (ACL).
22.9.2021 and 24.9.2021 Ch. 1: Introduction and motivation.
Ch. 2: The language of categories and functors (ACL Appendix A.3)
Chapter 1 notes
Chapter 2 notes
30.9.2021 and 1.10.2021 Ch. 2: Yoneda's Lemma (ACL Appendix A.3).
Ch. 3: Constructions of rings: quotients, polynomial rings, localization (ACL 1.5, 1.3 and 1.6).
Chapter 2 notes
Chapter 3 notes
6.10.2021 and 8.10.2021 Ch. 3: Localization; local rings, the Jacobson radical and Nakayama's Lemma (ACL 1.6, 2.1 and 6.1).
Ch. 4: Noetherian rings and modules (ACL 6.3).
Chapter 3 notes (updated)
Chapter 4 notes
13.10.2021 and 15.10.2021 Ch. 4: Noetherian rings and modules (ACL 6.3).
Ch. 5: the tensor product: definition, existence and functoriality (ACL 8).
Chapter 4 notes
Chapter 5 notes
20.10.2021 and 22.10.2021 Ch. 5: the tensor product (ACL 8): direct sums, free modules, commutativity and associativity, exactness.
Chapter 5 notes
27.10.2021 and 29.10.2021 Ch. 5: the tensor product (ACL 8), continuation: base change for modules and examples.
Chapter 5 notes
3.11.2021 and 5.11.2021 Ch. 5: the tensor product (ACL 8), continuation end end: tensor product of algebras, examples; multilinear algebra.
Chapter 5 notes
10.11.2021 and 12.11.2021 Ch. 6: dimension theory, I: definition and simple examples.
Chapter 6 notes
Ch. 7: integrality: basic definitions and properties.
Chapter 7 notes
17.11.2021 and 19.11.2021 Ch. 7: integrality: preservation of dimension and height; Cohen-Seidenberg's "going up" and "going down" theorems.
Chapter 7 notes
Ch. 8: Artinian rings and modules; simple modules.
Chapter 8 notes
24.11.2021 and 26.11.2021 Ch. 8: Artinian rings and modules; composition series and the Jordan-Hölder Theorem. Artinian rings and modules; Akizuki's theorem.
Chapter 8 notes
1.12.2021 and 3.12.2021 Ch. 8: the Hauptidealsatz, characterizations of UFDs.
Chapter 8 notes
Ch. 9: finitely-generated algebras over fields; transcendence degree, Noether's normalization theorem.
Chapter 9 notes
8.12.2021 and 10.12.2021 Ch. 9: Zariski's Theorem and the Nullstellensatz; height and dimension for finitely-generated algebras over fields.
Chapter 9 notes
Ch. 10: Discrete valuation rings.
Chapter 10 notes
15.12.2021 and 17.12.2021 Ch. 10: Discrete valuation rings: properties, characterizations among valuation rings and among rings.
Chapter 10 notes
22.12.2021 Ch. 10: completion of discrete valuation rings; the Lerch--Mahler--Skolem Theorem as example of application.
Chapter 10 notes


Primary Reference:
  1. "(Mostly) Commutative Algebra" by Antoine Chambert-Loir, denoted ACL.
Secondary Reference:
  1. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969)
Tertiary References:
  1. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989)
  2. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995)
  3. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer)