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Seminar on algebraic geometry at Bois Marie 1960-61. Étale coverings and fundamental group (SGA 1). A seminar directed by Alexander Grothendieck. Enlarged by two reports of M. Raynaud. Édition recomposée et annotée du original publié en 1971 par Springer. (Séminaire de géométrie algébrique du Bois Marie 1960-61. Revêtements étales et groupe fondamental (SGA 1). Un séminaire dirigé par Alexander Grothendieck. Augmenté de deux exposés de M. Raynaud.) (French) Zbl 1039.14001
Documents Mathématiques 3. Paris: Société Mathématique de France (ISBN 2-85629-141-4/hbk). xviii, 325 p. (2003).
This is the first of the famous series of “Séminaire de géométrie algébrique” of Alexander Grothendieck and his collaborators, started in 1960-61, in which Grothendieck constructs (and studies the main properties of) the algebraic fundamental group $$\pi^{\text{alg}}_1(X)$$ of a scheme $$X$$. If $$X$$ is a complex algebraic variety then the profinite completion of the topological fundamental group $$\pi_1(X)$$ coincides with $$\pi^{\text{alg}}_1(X)$$. This fundamental construction is extremely important when for example $$X$$ is an algebraic scheme over a finite field extension $$K$$ of $$\mathbb Q$$ because the knowledge of $$\pi^{\text{alg}}_1(X)$$ contains already a lot of information about $$X$$ (in some cases, enough to reconstruct $$X$$ itself). To quote Serre (from his talk in the Bourbaki Seminar of the fall of 1991), this seminar was the first great success of the theory of schemes.
Due to these facts it certainly deserved to be republished by the Société Mathématique de France as a volume of “Documents Mathématiques”. The present volume is a (slightly) corrected and updated version of the previous edition [Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. Some updated remarks have been added by M. Raynaud, which are bounded by brackets [ ] and indicated by the symbol (MR).

##### MSC:
 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc. 14E20 Coverings in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies 14F35 Homotopy theory and fundamental groups in algebraic geometry