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