Introduction in Sheaves and Schemes

Teaching semester

Irregular. Typically the course will be offered in the same semester as MAT229

Objectives and Content

Objectives:

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.

Contents:

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.

Learning Outcomes

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• is able to define and use fundamental notions and constructions and knows important results in algebraic geometry connected to sheaves and schemes
• 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.

Skills

The student

• masters fundamental techniques within sheaf and scheme theory
• is able to argue mathematically correct and present proofs and reasoning
• has solid experience and training in reasoning with sheaves and geometric stuctures

General competence

The student

• 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

MAT224 Commutative Algebra and MAT229 Algebraic Geometri I. MAT229 can be taken toghether with MAT320.

Compulsory Assignments and Attendance

No mandatory exercises or lectures

Forms of Assessment

Oral examination

Grading Scale

The grading scale used is A to F. Grade A is the highest passing grade in the grading scale, grade F is a fail.

Subject Overlap

MAT321: 5 ECTS