This course is an introduction to commutative algebra, i.e. we study commutative rings and modules over such rings. Basically, commutative algebra is algebraic geometry locally and therefore many ideas are motivated by geometry (sometimes vice versa). I will give you a brief idea of this connection but primarily focus on the algebra side. I will cover topics like rings, ideals, modules, prime ideals, spectrum, localization, chain conditions, dimension theory, primary decomposition, integrality, Dedekind domains. This course is extremely useful in basically all parts of algebra like algebraic geometry, algebraic number theory, computer algebra, and representation theory.
The course is virtual, managed through OLAT. There, you'll also find the videos for each week.
I have written up lecture notes (v0.1 for this course).
|01||26.10.-01.11.||1.1 – 1.12||Review of basic ring theory||Here|
|02||02.11.-08.11.||1.12 – 2.3 + first go on 2.4||Prime ideals: the basics||Here|
|03||09.11.-15.11.||Second go on 2.4 + 2.5 – 2.7||Prime ideals: geometry and topology||Here|
|04||16.11.-22.11.||3.1 – 3.4||Modules||Here|
|05||23.11.-29.11.||3.5 – 3.8||Homological stuff||Here|
|06||30.11.-06.12.||4.1 – 4.3.7||Localization||Here|
|07||07.12.-13.12.||4.3.7 – 5.1||Local properties, integrality||Here|
|08||14.12.-20.12.||5.2–5.4 + 6||Integrality, Nullstellensatz||Here|
|10||11.01.-17.01.||8.1 – 8.2||Dimension theory: basics||Here|
|11||18.01.-24.01.||8.3 – 8.4||Transcendence degree, codimension||Here|
|12||25.01.-31.01.||8.5 – 8.7||Krull PIT, regularity||Here|
Here's an (unfiltered) list of textbooks on the basics of commutative algebra:
- M. Atiyah and I. Macdonald, Introduction to commutative algebra. Addison-Wesley Publishing Co., 1969.
- N. Bourbaki, Commutative algebra. Chapters 1–7. Elements of Mathematics. Springer-Verlag, 1998.
- D. Eisenbud, Commutative Algebra. Graduate Texts in Mathematics, 150. Springer-Verlag, 1995.
- H. Matsumura, Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8. Cambridge University Press, 1989.
- M. Nagata, Local rings. Corrected reprint. Robert E. Krieger Publishing Co., Huntington, 1975.
- R. Sharp, Steps in commutative algebra. Cambridge University Press, 1990.
In my opinion, the best references at the beginning are the book by Atiyah–Macdonald (very concise, sometimes too concise, but still contains almost everything we need) and the book by Eisenbud (very detailed with lots of comments on geometric connections, probably too much at the beginning). The book by Bourbaki is a useful reference once you understood everything and have an intuition for it.