# Equivalence of categories between finite étale covers of connected scheme and finite continuous permutation representations of étale fundamental group

The title and tags say it all: I am looking for a clean statement and proof of the equivalence of categories between finite étale covers of a connected $$k$$-scheme and finite continuous permutation representations of the étale fundamental group $$\pi_1(X, \overline{x})$$ given by the fiber functor.

I need this result for my thesis, but I do not have the space necessary to prove it, so I would like a clear reference I can cite. This is stated in passing at the start of Section 6 in Szamuely's Galois Group and Fundamental Groups, but the proof consists of various remarks spread out throughout the entirety of the preceding section, and even then there's some work to be done. If possible, I would love a a wholly contained proof so that I can reference this away. I only need the result for reductive group schemes over the algebraic closure of a finite field, so total generality is not necessary.

• This is the content of SGA 1, exp. V. In particular Theorem 4.1, and §7 which explains that the theorem applies to connected schemes.
– abx
Apr 30, 2021 at 5:36
• @abx What exactly is SGA 1, exp. V? Apologies, I'm a novice in the field. Apr 30, 2021 at 5:48
• @mskilleter SGA is "Séminaire de Géometrie Algébrique" Apr 30, 2021 at 6:02
• SGA 1 Apr 30, 2021 at 8:26