BIREP — Representations of finite dimensional algebras at Bielefeld

• BIREP
• Faculty of Mathematics
• Bielefeld University

 

Commutative Algebra (WiSe 25-26)(ekvv 240151)

Monday 10-12 h (V5-148) and Wednesday 14-16 h in X-E0-208 (4 SWS)
Lecturer: Dr. Julia Sauter
Content: Commutative Algebra
Commutative algebra is the study of commutative rings and their modules. Usually after establishing ring theoretic properties we look at implications/applications for modules. This lecture follows a Bourbaki-style curriculum (summarized and modernized in [AK]) but includes also computational examples for polynomial rings over fields (following [CLO]). The highlight of this course is Hilbert's Nullstellensatz which is also the first step into Algebraic Geometry. But we will not circulate entirely around this result (as some courses do), our focus will be broader ensuring that you learn important technologies which also applied to /generalized to / inspired further concepts in category theory (such as localization, Hom-tensor adjunction, completion and even Groebner basis theory).
This is a tentative list of chapters for this course - rings will always assumed to be commutative.

• Rings and ideals
• Prime ideals
• radical ideals
• Hilbert's Nullstellensatz for polynomial rings over algebraically closed fields
• Modules and exact sequences
• Some category theory
• Tensor products
• Flatness
• Lemma of Nakayama
• Localization
• Support
• Going-up and down
• Noether normalization and Hilbert's Nullstellensatz
• Chain conditions
• Associated primes
• Primary decomposition
• Finite length
• Dimension
• Graded rings and modules
• Projective Nullstellensatz for polynomial rings

In the Lernraum you will find further information to this course.
Literature: Main literature is [AK], some material we take from [CLO].

• [AK] Altman-Kleiman: A term in commutative algebra (see here)
• [CLO] Cox-Little-O'Shea: Ideals, Varieties and Algorithms (in the bib)

Tutorium: Thurday 18-20h in ... by Andrew Hubery
This course will have weekly homework sheets. They contain two types of exercises: 1) A number X of "presence exercises" which vary in length and difficulty and 2) precisely 2 "hand-in exercises".
Presence exercises will be discussed in the tutorium by you on the blackboard with guidance of the tutor. According to the participants it may happen that not all of these are discussed or that you discuss instead a question of you or that the tutor adds spotaneously some more exercises.
The two hand-in exercises will be marked and returned to you by the tutor - you can work together with at most one partner (both hand writings must be recognizable). It is optional (depending on the students) to also discuss their solution in the turorium when the sheets are returned.
To get the Leistungsnachweis for the tutorium you have to actively and regularly participate in the tutorial and gain half of the total number of points for the hand-in exercises. You should be aware that "presence exercises" still require some thoughts of you before the tutorium. Note down these thoughts and questions and come prepared then it is much more fun and interesting for you.