11.1 Étale morphisms

Étale morphisms are a key concept in arithmetic geometry, bridging algebraic geometry and number theory. They generalize local isomorphisms from complex analysis to algebraic varieties, providing insights into scheme structure and cohomological properties.

These morphisms exhibit local ringed space isomorphisms and satisfy formal smoothness and unramified conditions. They're flat, finitely presented, and have zero-dimensional fibers. Understanding étale morphisms is crucial for applications in Galois theory and cohomology.

Definition of étale morphisms

• Étale morphisms form a crucial concept in arithmetic geometry, bridging algebraic geometry and number theory
• These morphisms generalize the notion of local isomorphisms in complex analysis to algebraic varieties
• Understanding étale morphisms provides insights into the structure of schemes and their cohomological properties

Local structure

• Étale morphisms exhibit locally ringed space isomorphisms at each point
• Induce isomorphisms between completed local rings $\hat{\mathcal{O}}_{Y,f(x)} \cong \hat{\mathcal{O}}_{X,x}$ for every point x in X
• Preserve dimension and regularity of local rings
• Behave like "algebraic local diffeomorphisms" in the context of schemes

Formal smoothness condition

• Étale morphisms satisfy the infinitesimal lifting property
• For any affine scheme T and a nilpotent ideal I in O_T, any T/I-valued point of X lifts uniquely to a T-valued point
• Ensures the morphism is formally smooth in the sense of deformation theory
• Allows for the extension of morphisms along infinitesimal thickenings

Unramified condition

• Étale morphisms are unramified, meaning they have trivial relative cotangent sheaf
• Cotangent sheaf $\Omega_{X/Y} = 0$ for an étale morphism f: X → Y
• Implies the morphism is injective on tangent spaces at each point
• Guarantees that the fibers of the morphism are discrete

Properties of étale morphisms

• Étale morphisms possess unique characteristics that make them essential in arithmetic geometry
• These properties allow for the transfer of information between schemes and their étale covers
• Understanding these properties is crucial for applications in Galois theory and cohomology

Flatness and finite presentation

• Étale morphisms are always flat, ensuring good behavior of fibers
• Flatness implies that the morphism preserves dimension and depth of local rings
• Finite presentation guarantees that the morphism is "locally of finite type" and quasi-compact
• Allows for the use of Noetherian approximation techniques in studying étale morphisms

Fiber dimension

• Fibers of étale morphisms are discrete and have dimension zero
• Each fiber consists of a finite number of points (finite étale case)
• Points in the fiber correspond to separable field extensions of the residue field at the image point
• Fiber dimension property distinguishes étale morphisms from more general smooth morphisms

Base change stability

• Étale morphisms are stable under base change, preserving their properties
• For an étale morphism f: X → Y and any morphism Y' → Y, the pullback X ×_Y Y' → Y' is also étale
• Allows for the study of étale morphisms in different geometric contexts
• Crucial for defining étale topology and sheaf theory on schemes

Étale vs smooth morphisms

• Étale and smooth morphisms share some properties but differ in crucial aspects
• Both concepts generalize the notion of submersions from differential geometry to algebraic geometry
• Understanding their relationship is essential for applications in arithmetic geometry

Similarities and differences

• Both étale and smooth morphisms are formally smooth and locally of finite presentation
• Smooth morphisms allow for positive fiber dimension, while étale morphisms have zero-dimensional fibers
• Étale morphisms can be thought of as "smooth morphisms of relative dimension zero"
• Smooth morphisms satisfy the infinitesimal lifting property for all Artinian rings, not just nilpotent thickenings

Examples and counterexamples

• The projection $\mathbb{A}^2_k \to \mathbb{A}^1_k$ is smooth but not étale (positive fiber dimension)
• The inclusion of a point $Spec(k) \to \mathbb{A}^1_k$ is neither smooth nor étale (not flat)
• The Frobenius morphism in characteristic p is étale for perfect fields but not smooth
• The normalization of a nodal curve is an example of a finite morphism that is generically étale but not étale

Étale topology

• Étale topology provides a finer notion of "locality" than the Zariski topology
• This topology is fundamental in modern arithmetic geometry and algebraic number theory
• Allows for the study of cohomological properties that are invisible in the Zariski topology

Étale coverings

• An étale covering of a scheme X is a family of étale morphisms {U_i → X} that are jointly surjective
• Étale coverings generalize open coverings in the Zariski topology
• Allow for the consideration of field extensions and Galois theory in a geometric context
• Provide a framework for studying local properties that are not detectable in the Zariski topology

Étale sites

• The étale site of a scheme X consists of all étale morphisms U → X
• Forms a category with a Grothendieck topology defined by étale coverings
• Allows for the definition of sheaves and cohomology theories more sensitive than Zariski cohomology
• Crucial for the formulation of étale cohomology and the study of l-adic sheaves

Sheaves in étale topology

• Sheaves on the étale site capture more information than Zariski sheaves
• Include important examples like the sheaf of n-th roots of unity μ_n
• Allow for the definition of étale cohomology groups H^i(X_ét, F) for étale sheaves F
• Provide a framework for studying Galois representations and arithmetic properties of schemes

Applications in arithmetic geometry

• Étale morphisms play a central role in modern arithmetic geometry
• They provide a bridge between algebraic geometry and number theory
• Applications of étale theory have led to significant advances in the field

Galois theory connection

• Étale morphisms generalize the notion of Galois extensions to schemes
• Finite étale covers correspond to finite separable extensions in the function field case
• Allow for the formulation of the fundamental group of schemes, generalizing Galois groups
• Provide a geometric interpretation of ramification and splitting of primes in number fields

Étale cohomology

• Étale cohomology theory, developed by Grothendieck, revolutionized arithmetic geometry
• Provides a cohomology theory for schemes that behaves like singular cohomology for complex varieties
• Crucial for the proof of the Weil conjectures by Deligne
• Allows for the study of l-adic representations and their connections to automorphic forms

Fundamental groups

• The étale fundamental group generalizes both Galois groups and topological fundamental groups
• For a connected scheme X, π_1^ét(X) classifies finite étale covers of X
• Provides a unifying framework for studying Galois theory and covering space theory
• Allows for the formulation of anabelian geometry and Grothendieck's section conjecture