Commutative Algebra Autumn 2021

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

Content

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:

• The language of categories and functors
• Basic facts about rings, ideals and modules
• Constructions of rings: quotients, polynomial rings, localization
• Noetherian rings and modules
• The tensor product of modules over commutative rings and its applications
• Krull dimension
• Integral extensions and the Cohen-Seidenberg theorems
• Finitely generated algebras over fields, including the Noether Normalization Theorem and the Nullstellensatz
• Artinian rings and modules
• Primary decomposition
• Discrete valuation rings and some applications

Prerequisites

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).

Forum

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

Exercises

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 classes

Content of the Lectures

Literature

1. "(Mostly) Commutative Algebra" by Antoine Chambert-Loir, denoted ACL.

2. "Introduction to Commutative Algebra" by M. F. Atiyah and I. G. Macdonald (Addison-Wesley Publ., 1969)

4. "Commutative ring theory" by H. Matsumura (Cambridge University Press 1989)
5. "Commutative algebra. With a view towards algebraic geometry" by D. Eisenbud (GTM 150, Springer Verlag, 1995)
6. "Commutative Algebra" by N. Bourbaki (Hermann, Masson, Springer)