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# Algebra 2 (Kommutative Algebra)

## Module MA5120

This Module is offered by Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of SS 2012

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2021/2WS 2020/1SS 2020SS 2019SS 2012

### Basic Information

MA5120 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

• Catalogue of non-physics elective courses
Total workloadContact hoursCredits (ECTS)
270 h 90 h 9 CP

### Content, Learning Outcome and Preconditions

#### Content

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

#### Learning Outcome

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.

#### Preconditions

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

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

TypeSWSTitleLecturer(s)DatesLinks
VO 4 Algebra 2 [MA5120] Kemper, G. Tue, 10:15–11:45, MI 00.07.014
Thu, 10:15–11:45, MI 00.07.014
and singular or moved dates
eLearning

#### Learning and Teaching Methods

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.

#### Media

blackboard, exercise sheets

#### Literature

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.

### Module Exam

#### Description of exams and course work

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.

#### Exam Repetition

The exam may be repeated at the end of the semester.

#### Current exam dates

Currently TUMonline lists the following exam dates. In addition to the general information above please refer to the current information given during the course.

Title
TimeLocationInfoRegistration
Algebra 2
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