Fundamental groups in arithmetic and algebraic geometry
TCC course, Fridays 2-4, starting October 12.
Lecture notes (only up to lecture 5-ish, questions, comments and corrections very welcome!)
This course will be focused on the definition and basic properties of the etale fundamental group. The course will assume some knowledge of algebraic geometry (e.g. Harthshorne chapter two) but will be sufficiently leisurely that it should be accessible if that is knowledge you are in the process of acquiring. The rough plan of the course is as follows:
Etale morphisms - definitions
Galois categories
The Riemann existence theorem
Galois groups and fundamental groups, the homotopy exact sequence
The specialisation theorem for the etale fundamental group
References:
SGA 1
A. Cadoret, Galois categories
J.P. Murre, Lectures on an introduction to Grothendieck's theory of the fundamental group
J. Milne, Etale Cohomology
Stacks Project
T. Szamuely, Galois groups and fundamental groups
EGA IV.4
H. Lenstra, Galois theory for schemes
L. Illusie, Grothendieck's existence theorem in formal geometry, in "FGA Explained"
M. Hadian, Lectures on etale fundamental groups