Lecturer

Dr. Massimiliano Alessandro
Office hours: by appointment via email.

Tutor

Tobias Schnieders
Office hours: by appointment via email.

Time and Place

Lectures: Tuesday 16-18 in SR6, Wednesday 14-16 in SR6.
Exercise Class: Thursday 14-16 in SR6.

Note: The course will be held in English.

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)

Registration

Please enroll in the course via the CMS website.

Exam Dates

First Exam (Hauptklausur): 30.07.2026, 9-12, in HS IV (Gebäude E2 4), Zeichensaal (U.39, 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.

Exam Criteria

Exercise sheets are uploaded weekly (on Tuesdays) on the CMS website and must be solved by the following week (Tuesday). Submission of the exercise sheets takes place via CMS. The submitted sheets will be corrected and returned with the awarded points via CMS.

In order to obtain a grade, the following criteria are to be fulfilled:

• regular participation in the exercise classes,


• at least 50% of the points from the exercise sheets attained,


• the passed exam.

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 set up a course reserve for this lecture, see here.