Algebra 2 (Kommutative Algebra)

- Basic algebraic concepts for commutative algebra (Rings, ideals, prime ideals, modules)
- Localization
- Valuations
- Tensor products
- Affine algebraic varieties
- Finiteness conditions, dimension theory

At the end of the module students understand basic concepts of commutative algebra and their relation to the theory of affine varieties. They are able to apply the learned theorems and methods to particular problems concerning rings, in particular finitely generated algebras over a field, and translate them into results on the corresponding affine varieties.

MA1101 Linear Algebra and Discrete Structures 1, MA1102 Linear Algebra and Discrete Structures 2, MA2101 Algebra
Bachelor 2019: MA0004 Linear Algebra 1, MA0005 Linear Algebra 2 and Discrete Structures, MA2010 Algebra

lecture and exercise course, homework, assignments
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.

blackboard, exercise sheets

Atiyah/Macdonald: Introduction to commutative algebra, Addison-Wesley Publishing Co.
Eisenbud: Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York.

The exam will be in written (90 min) or oral (25 min) form, depending on the number of participants. Students demonstrate that they have gained deeper knowledge of definitions, proofs, and main mathematical tools and results in commutative algebra, and that they understand their relation to basic concepts in algebraic geometry. The students are expected to be able to explain the methods and derive the main results, to explain their properties, and to apply them to specific problems and examples.