Objectives and Content
The course aims at giving a first introduction to sheaves and schemes in
algebraic geometry and their fundamental properties. This forms the
basis of modern algebraic geometry.
The subject studies sheaves and morphisms between them,
especially short exact sequences, and definition and fundamental
properties of locally ringed spaces and schemes and morphisms between
schemes. In particular, the notions affine, noetherian, integral,
reduced, irreducible, separated, proper and projective schemes are
considered, as well as the structure sheaf, generic point, closed and
open embeddings, fiber product and fiber. The connection between schemes
and varieties is also studied.
On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:
* is able to define and use fundamental notions and constructions and
knows important results in algebraic geometry connected to sheaves and
* is able to perform a simple analysis of schemes, in particular using
properties of well-known sheaves.
* is able to produce the main ideas in the proofs of the most important
results connected to the notions above.
* masters fundamental techniques within sheaf and scheme theory
* is able to argue mathematically correct and present proofs and
* has solid experience and training in reasoning with sheaves and
* is able to work individually and in groups
* is able to formulate in a precise and scientifically correct way
* is able to decide whether complex mathematical arguments are correct
Recommended Previous Knowledge
Compulsory Assignments and Attendance
No mandatory exercises or lectures
Forms of Assessment
The grading scale used is A to F. Grade A is the highest passing grade in the grading scale, grade F is a fail.
MAT321: 5 ECTS
Type of assessment: Oral examination
- Withdrawal deadline