# The Étale Fundamental Group

Seminar organised by Jinbi Jin (jinmengjun [plus] efg [at] gmail [dot] com)

## Time and location

In week 51, the seminar will be on Monday, 15:00-17:00, instead of the usual time.

Wednesday, 11:15-12:50, Room Sn401

## Outline

This seminar will have two main goals. The first is to define the étale fundamental group of a connected scheme $$S$$ using a categorical approach, and the second is to prove a finiteness theorem about étale fundamental groups of proper, connected schemes over an algebraically closed field $$k$$. For these purposes, the following topics will be covered.

### Part I: Definitions

• Galois categories
• Finite étale morphisms of schemes
• Fibre functors; the étale fundamental group

### Part II: Finiteness

• Descent
• GAGA (Géométrie Algébrique et Géométrie Analytique); Riemann Existence Theorem
• Specialisation theory for the étale fundamental group
• Chow's Lemma
• Bertini's Theorem

## Schedule

 2 October Jinbi Jin Introduction 9 October Chloe Martindale Galois Categories 16 October Maxim Mornev Flat and Étale Morphisms I 23 October Maxim Mornev Flat and Étale Morphisms II 30 October Milan Lopuhaä The Galois Category of Finite Étale Coverings 6 November Wouter Zomervrucht Faithfully Flat Descent 13 November Qijun Yan Descent of Properties of Morphisms→ leidenuniv.nl 20 November --- No seminar. 27 November Yan Zhao GAGA 4 December Yan Zhao Riemann Existence Theorem 11 December Jinbi Jin Specialisation Theory 16 December, 15:00-17:00 Qijun Yan ''Finiteness'' of the Étale Fundamental Group