SGA — English Translation

SGA 1: Étale Coverings and the Fundamental Group

Séminaire de Géométrie Algébrique du Bois Marie, 1960-61.

A seminar directed by A. Grothendieck, augmented by two exposés by Mme M. Raynaud.

Abstract

This volume is a retypeset and annotated edition of Revêtements Étales et Groupe Fondamental, Lecture Notes in Mathematics 224, Springer-Verlag, Berlin-Heidelberg-New York, 1971, by Alexander Grothendieck et al.

The text presents the foundations of a theory of the fundamental group in algebraic geometry, from the “Kroneckerian” point of view that makes it possible to treat on the same footing the case of an algebraic variety in the usual sense and, for example, that of the ring of integers of a number field.

Subject Class

14-02, 14A15, 14B25, 14D15, 14E20, 14F35, 14H30

Keywords

Étale morphism, smooth morphism, flat morphism, scheme, fundamental group, covering, descent theory, specialization.

Preface

The present text is a retypeset and annotated edition of Revêtements Étales et Groupe Fondamental, Lecture Notes in Mathematics 224, Springer-Verlag, Berlin-Heidelberg-New York, 1971, by Alexander Grothendieck et al.

The composition in LaTeX 2e was carried out by volunteers participating in a project directed by Bas Edixhoven; more details can be found at http://www.math.leidenuniv.nl/~edix/. The page layout was completed by the Société mathématique de France.

This is a slightly corrected version of the original text. It is published by the Société mathématique de France in the series Documents Mathématiques. Some updates were made by Michel Raynaud. They are delimited by brackets and marked by the symbol (MR): remarks on pages X.2.14, XI.1.4, XII.5.6, XIII.2.13, and the note III.6.6.p24. To make the notation uniform, the residue field of a point x is denoted κ(x), and the residue field of a local ring A is denoted κ(A).

There also exists an electronic version intended to reproduce the original text.

The two versions are produced from a single source file, located on the arXiv.org e-print server at $<http://arxiv.org/$>. The differences between the two versions are documented in the source file.

The old page numbering is incorporated in the margin; the number n indicates the beginning of page n.

Contents of this volume

• Foreword
• Introduction
• Exposé I. Étale Morphisms
• Exposé II. Smooth Morphisms: Generalities, Differential Properties
• Exposé III. Smooth Morphisms: Extension Properties
• Exposé IV. Flat Morphisms
• Exposé V. The Fundamental Group: Generalities
• Exposé VI. Fibered Categories and Descent
• Exposé VII. Does Not Exist
• Exposé VIII. Faithfully Flat Descent
• Exposé IX. Descent of Étale Morphisms. Application to the Fundamental Group
• Exposé X. Theory of Specialization of the Fundamental Group
• Exposé XI. Examples and Complements
• Exposé XII. Algebraic Geometry and Analytic Geometry
• Exposé XIII. Cohomological Properness of Sheaves of Sets and of Sheaves of Noncommutative Groups
• Translation glossary
• Index of notations