Étale Fundamental Groups and Cohomology
Over a field, an étale scheme can be recovered from its geometric fiber together with the action of the field's Galois group. For a general scheme, finite étale covers are likewise described by finite locally constant sheaves and, on a connected base with a geometric point, by fundamental-group actions. The results explain when changes to a scheme preserve its covers and how its fundamental group combines geometry with the arithmetic of the base field.
The cohomological results describe what can be recovered from local data. The Kummer sequence records the obstruction to extracting a global root, while Hilbert 90 expresses the triviality of line bundles over a field; the latter theorem here assumes characteristic zero. The proper base change result shows that sections on a proper scheme over a henselian local ring are determined by its closed fiber. This comparison concerns global sections, or degree-zero cohomology.
Selected theorems and definitions
- definition
Étale schemes over a field as continuous Galois sets
The geometric fiber over a separable closure gives an equivalence for all étale schemes over a field, including those with infinitely many components.
- theorem
Uniqueness up to isomorphism of a finite-action fiber functor
An exact finite-set-valued functor on finite continuous group actions is isomorphic to the forgetful functor.
- theorem
Five criteria for surjectivity on fundamental groups
A morphism of fiber-functor data is surjective on automorphism groups precisely when the corresponding functor preserves connectedness or satisfies equivalent categorical conditions.
- theorem
The geometric–arithmetic exact sequence
For a geometrically connected, quasi-compact, quasi-separated scheme over a field, the geometric fundamental group is the kernel of the arithmetic Galois action.
- theorem
Finite étale covers are invariant under universal homeomorphisms
An integral, universally injective, surjective morphism induces an equivalence of finite étale cover categories and hence the expected pointed fundamental-group isomorphism.
- theorem
The Kummer sequence on the small étale site
When a positive integer is invertible on the scheme, taking its power on the multiplicative group is locally surjective in the étale topology.
- theorem
Vanishing of derived étale H¹ of the multiplicative group over a field
The first étale cohomology of the multiplicative group vanishes over a characteristic-zero field.
- definition
Finite étale schemes and finite locally constant sheaves
The sheaf of sections identifies finite étale schemes with finite locally constant sheaves of sets on the small étale site.
- theorem
Exact direct image for a closed immersion
Direct image along a closed immersion preserves short exact sequences of étale sheaves of modules.
- theorem
Degree-zero proper base change over a henselian local ring
A section of an étale module sheaf on a proper scheme over a henselian local ring is uniquely determined by its restriction to the closed fiber, and every closed-fiber section lifts.
References used in these guides
- A. Grothendieck (director), with two exposés by M. Raynaud. Revêtements étales et groupe fondamental (SGA 1). Séminaire de Géométrie Algébrique du Bois Marie 1960–61. Société Mathématique de France, Documents Mathématiques 3, xviii + 327, 2003. Revised and annotated edition of the seminar volume first published in 1971 as Lecture Notes in Mathematics 224.
- The Stacks Project Authors. Lemma 58.2.2: Schemes étale over a point. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Lemma 58.3.4: Fiber functors on finite continuous group actions. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Lemma 58.4.1: Functors and homomorphisms of fundamental groups. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Lemma 58.14.3: Geometric and arithmetic fundamental groups. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Proposition 58.8.4: Topological invariance of finite étale covers. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Section 59.28: Kummer theory. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Section 59.24: Picard groups. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Section 59.64: Locally constant sheaves. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Section 59.46: Closed immersions and pushforward. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.
- The Stacks Project Authors. Section 59.91: The proper base change theorem. 2026. Living reference; consulted 15 October 2026. The tag is the stable identifier.