L'Étale

L'Étale. In algebraic geometry, an L'Étale morphism is a fundamental concept defining a type of map between schemes that is formally analogous to a local isomorphism in differential geometry. It serves as the cornerstone for defining étale cohomology, a pivotal Weil cohomology theory used to prove the Weil conjectures. The study of these morphisms is central to the Grothendieck's revolution in the field, linking geometric intuition with sophisticated algebraic-topological methods.

Definition and mathematical context

Formally, a morphism of schemes \( f: X \to Y \) is called L'Étale if it is flat and unramified. The flatness condition ensures nice behavior under base change, while being unramified means its diagonal morphism is an open immersion. In the affine case, this corresponds to the ring map making the source scheme a formally étale algebra over the target scheme. This definition was solidified in the work of Alexander Grothendieck within the Éléments de géométrie algébrique and further developed by Michael Artin and Jean-Louis Verdier. The concept generalizes the idea of a covering space from topology to the setting of schemes, where the Zariski topology is too coarse, necessitating the étale topology.

Properties and examples

Key properties include being locally of finite presentation and smooth of relative dimension zero. Étale morphisms are stable under composition, base change, and fiber products. A fundamental example is the map from the affine line to itself via \( x \mapsto x^n \) away from zero in characteristic not dividing \( n \), which is étale. The inclusion of a Zariski open subset is étale, as is any isomorphism of schemes. In number theory, the map \(\text{Spec}(\mathbb{Q}(i)) \to \text{Spec}(\mathbb{Q})\) induced by the field extension is étale. Over the complex numbers, étale morphisms correspond to local homeomorphisms in the analytic topology under the GAGA principle.

Relation to other morphisms

The class of étale morphisms sits strictly between smooth morphisms and unramified morphisms. Every étale morphism is smooth, but not every smooth map (like \(\mathbb{A}^1 \to \mathbb{A}^1, x \mapsto x^2\) in characteristic 2) is étale. It is also finer than being flat; a flat morphism like the projection of a nodal curve can fail to be étale. Étale maps are closely related to finite étale covers, which are analogous to finite covering spaces. The theory connects deeply with Galois theory through the étale fundamental group, developed by Grothendieck in his SGA 1.

Applications in algebraic geometry

The primary application is the construction of étale cohomology, a cohomology theory for schemes that behaves like singular cohomology for complex varieties. This theory was essential to Pierre Deligne's proof of the Weil conjectures, particularly the Riemann hypothesis for varieties over finite fields. Étale cohomology provides the framework for defining ℓ-adic cohomology and lisse sheaves. It is also crucial in the study of abelian varieties via Tate modules and in the development of Grothendieck's anabelian geometry. The étale site forms a Grothendieck topology key to topos theory.

Generalizations and related concepts

Significant generalizations include the notion of a formally étale morphism in category theory and algebraic geometry. The concept extends to stacks and higher category theory, leading to derived algebraic geometry. Related structures include Nisnevich morphisms, which are étale morphisms with a stronger condition on rational points, and fppf morphisms. In arithmetic geometry, crystalline cohomology and syntomic cohomology were developed as successors for contexts where étale cohomology fails, such as in positive characteristic. The study of perfectoid spaces by Peter Scholze heavily utilizes pro-étale morphisms and the pro-étale topology.

Category:Algebraic geometry Category:Morphisms of schemes Category:Étale cohomology