Abstract

In SGA1 Grothendieck generalized Galois theory for fields to arbitrary connected schemes. The "Galois group" in this generalized Galois theory is the étale fundamental group. However, as Bhatt-Scholze observed, this fundamental group does not behave quite well when the scheme is not normal. For example, there are l-adic local systems which do not come from any l-adic representation of the fundamental group. To remedy this, Bhatt-Scholze constructed the pro-étale fundamental group. This fundamental group takes the étale fundamental group as its profinite completion in general, but equals it when the scheme is normal. In this talk, I will first give a detailed introduction to the pro-étale fundamental group, then I will compare it with the topological fundamental group when the scheme is taken to be a complex algebraic variety. If time permits, I will also present some other aspects of this fundamental group such as the specialization.