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 |
time | room | assistant | language |
---|---|---|---|
Th 09-10 | HG E 1.2 | Younghan Bae | English |
Th 12-13 | HG E 1.2 | Ilaria Viglino | English |
Day | Content |
---|---|
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 |