Scheme Theory

Prerequisites:

Abstract Algebra II, Fundamental Algebra and Analysis II

Course description:

This course is designed for students interested in pursuing research in algebraic geometry or arithmetic geometry. The teaching objectives are to equip students with the foundational language and fundamental methods of scheme theory, preparing them for advanced studies in algebraic geometry and arithmetic geometry. Additionally, the course aims to cultivate students' innovative thinking and stimulate their interest in mathematics and mathematical research.

Learning objectives:

By the end of this semester-long course, students should achieve the following learning outcomes: They will acquire a solid understanding of the fundamental concepts in scheme theory and develop essential skills for studying algebraic geometry. Through the course, students will be introduced to key ideas in modern algebraic geometry and learn how to effectively search for and comprehend research literature in the field. Furthermore, they will gain initial hands-on experience in conducting algebraic geometry research and presenting their findings, providing them with valuable exposure to actual mathematical research practices.

Detailed topics covered:

1. Basic properties of affine scheme: as sets, topological spaces and functors

2. Basic of sheaf theory: the structure sheaf of schemes, and basic properties of schemes

3. Global properties of morphisms: separated, proper, projective and finite morphisms

4. Coherent sheaves on schemes: quasi-coherent and coherent sheaves, direct and inverse image

5. Local properties of morphisms: flat and smooth morphisms

6. Divisor on schemes: Weil and Cartier divisors, invertible sheaves

7. Cohomology of sheaves: Cech cohomology and cohomology of projective spaces

8. Application of cohomology: Riemann-Roch theorem on curves