I am looking for exact references for the comparison theorem for the étale fundamental group. I mean the following result:

Theorem (Grothendieck). For a pointed algebraic variety $(X,x)$ over $\mathbb{C}$ there is a canonical isomorphism between the étale fundamental group $\pi_1^{\text{ét}}(X,x)$ and the profinite completion of the topological fundamental group $\pi_1^{\rm top}(X(\mathbb{C}),x)$.

(I am interested in the case when $X$ is nonsingular.)

I could not find this assertion in SGA1 or in the book "Galois Groups and Fundamental Groups" by Tamás Szamuely. Please help!

  • 3
    $\begingroup$ This is Corollary 5.2 in SGA 1, Exposé XII. $\endgroup$
    – abx
    Feb 14, 2017 at 13:38
  • 1
    $\begingroup$ As everyone else pointed out, this is SGA1, Exposé XII, Cor 5.2. Since you also said that you couldn't find it on Szamuely's book I wanted to add that you can find it there as Thm 5.7.4. (I have not enough reputation to add this as a comment). $\endgroup$
    – Pippo
    Oct 11, 2019 at 11:25

2 Answers 2


This is in SGA1, Exposé XII, Section 5.


See also Artin-Mazur "Etale Homotopy", Ch. 12.


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