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Algebra 2 (Kommutative Algebra)

Modul MA5120

Dieses Modul wird durch Fakultät für Mathematik bereitgestellt.

Diese Modulbeschreibung enthält neben den eigentlichen Beschreibungen der Inhalte, Lernergebnisse, Lehr- und Lernmethoden und Prüfungsformen auch Verweise auf die aktuellen Lehrveranstaltungen und Termine für die Modulprüfung in den jeweiligen Abschnitten.

Basisdaten

MA5120 ist ein Semestermodul in Englisch auf Master-Niveau das im Sommersemester angeboten wird.

Das Modul ist Bestandteil der folgenden Kataloge in den Studienangeboten der Physik.

  • Allgemeiner Katalog der nichtphysikalischen Wahlfächer
GesamtaufwandPräsenzveranstaltungenUmfang (ECTS)
270 h 90 h 9 CP

Inhalte, Lernergebnisse und Voraussetzungen

Inhalt

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

Lernergebnisse

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.

Voraussetzungen

MA1101 Linear Algebra and Discrete Structures 1, MA1102 Linear Algebra and Discrete Structures 2, MA2101 Algebra

Lehrveranstaltungen, Lern- und Lehrmethoden und Literaturhinweise

Lehrveranstaltungen und Termine

ArtSWSTitelDozent(en)Termine
VO 4 Algebra 2 [MA5120] Kemper, G. Di, 10:15–11:45, 5607.EG.014
Do, 10:15–11:45, 5607.EG.014
sowie einzelne oder verschobene Termine

Lern- und Lehrmethoden

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.

Medienformen

blackboard, exercise sheets

Literatur

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.

Modulprüfung

Beschreibung der Prüfungs- und Studienleistungen

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.

Wiederholbarkeit

Eine Wiederholungsmöglichkeit wird am Semesterende angeboten.

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