Commutative Algebra and Introduction to Algebraic Geometry

Commutative Algebra and Introduction to Algebraic Geometry

 

Register for the course on this page!

Note: The course will be held in English.

 

Lecturer: Dr. Massimiliano Alessandro

Office hours: by appointment via email.

 

If a lecture takes place online, the recorded video lecture will be uploaded. Each video will be available online for two weeks and will then be deleted.

 

Lectures

Tuesday 16-18 in SR6, Wednesday 14-16 in SR6.

 

Learning Goals
After completing this course, students will be able to understand and apply the basic concepts, methods, and techniques of Commutative Algebra and classical Algebraic Geometry.

 

Content

• Hilbert's Basis theorem, Hilbert's Nullstellensatz
• modules, tensor products of modules
• localisation and local properties of rings and modules
• integral dependence, valuations, going-up and going-down theorems
• graded rings and modules
• affine varieties, projective varieties, Zariski topology
• morphisms and rational maps between varieties
• birational morphisms and blowups
• nonsingularity and smoothness

Prerequisites: Linear Algebra I and II, Algebra (recommended)

 

Workload

• 4 hours per week (lectures), 2 hours per week (exercise classes)

• 9 CP

• 60 hours of work for lectures

• 30 hours of work for exercise classes

• 180 hours of self-study (time for preparation and review of lectures and exercise classes, as well as completion of exercise sheets)

 

Exam Criteria

Exercise sheets will be published weekly (on Tuesdays) in CMS and must be solved by the following week (Tuesday). Please log in first. You will find the exercise sheets under the “Materials” section.

Submission of the exercise sheets takes place via CMS. Students may submit solutions in groups of at most three people. Please, create your own team under the "Team Groupings" section.

The submitted sheets will be corrected and returned with the awarded points via CMS.

In order to obtain a grade, students must meet all of the following requirements:

• regular participation in the exercise classes

• at least 50% of the total available points from the exercise sheets

• a passing grade on the final exam

 

Exercise Classes

During the exercise sessions, solutions to the exercise sheets are presented, and students have the opportunity to ask questions about the lecture material, the grading of the submitted exercise sheets, or any general issues related to the exercise sheets.

Exercise Class: Thursday 14-16 in SR6.

Tutor: Tobias Schnieders

Office hours: TBA.

 

Exam Dates

First Exam (Hauptklausur): 30.07.2026, 9-12, in HS II (Gebäude E2 5)
Second Exam (Nachklausur): 07.10.2026, 9-12, in HS III (Gebäude E2 5)

Important: If you are unable to register for the first or second exam in LSF, please contact me by email (your examination office must send me a confirmation in good time before the exam stating that you are permitted to take it). If permitted by your examination regulations, the second exam may be used to improve your grade.

 

Literature

• Atiyah, Macdonald. Introduction to Commutative Algebra.
• Matsumura. Commutative Algebra.
• Matsumura. Commutative Ring Theory.
• Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry.
• Hartshorne. Algebraic Geometry.
• Vakil. Foundations of Algebraic Geometry.
• Fulton. Algebraic Curves. An Introduction to Algebraic Geometry.
• Görtz, Wedhorn. Algebraic Geometry I.

The Campus-Bibliothek für Informatik und Mathematik has established a course reserve for this lecture.