The Étale Fundamental Group

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

References

A. Grothendieck, SGA I→ arXiv.org
H. W. Lenstra, Galois theory for schemes→ leidenuniv.nl

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