Étale fundamental group
Prerequisiti
The course is recommended for fourth or fifth year and Ph.D. students.
The minimum prerequisite is a basic course in algebraic geometry and one in topology in which the fundamental topological group has been covered. It is strongly recommended to have taken Scheme Theory I, and it is helpful to have taken Scheme Theory II, or otherwise have a knowledge of scheme theory at the level of Hartshorne's book.
Programma
The course will be divided into two parts: in the first part (which will last about two-thirds of the course) we will develop the theory of the étale fundamental group. In the second part we will address some selected applications and topics: I will make some proposals for this second part, but students are invited to propose alternative topics more of their interest.
Part 1.
- Unramified and étale morphisms.
- Category of étale coverings. Galois categories.
- The fundamental group of a Galois category. Morphisms of profinite groups and Galois categories.
- Galois groups as fundamental groups. Comparison with topological fundamental group. Fundamental group of a smooth curve.
- Exact homotopy sequences. Specialization of the fundamental group.
- If there is time, mention of Nori's fundamental group and Tannakian categories.
Part 2. Possible topics, to be determined according to students' interests. Different proposals are welcome.
- The tame fundamental group of a curve in positive characteristic.
- J. Stix "On the period-index problem in light of the section conjecture."
- A. Tamagawa "The Grothendieck conjecture for affine curves".
- Belyi's theorem and consequences for the Galois group of Q.
Obiettivi formativi
The purpose of the course is to introduce the étale fundamental group and see some applications.
Riferimenti bibliografici
- J. P. Murre, Lectures on an Introduction to Grothendieck’s Theory of the Fundamental Group.
- T. Szamuely, Galois groups and fundamental groups.
- A. Grothendieck et al.t, SGA 1.
- M. Nori, The fundamental group scheme.