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 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 O^Y,f(x)≅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 ΩX/Y=0 for an 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 , 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 Ak2→Ak1 is smooth but not étale (positive fiber dimension)
The inclusion of a point Spec(k)→Ak1 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 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
Key Terms to Review (16)
Alexander Grothendieck: Alexander Grothendieck was a groundbreaking mathematician known for his profound contributions to algebraic geometry, particularly in developing the modern foundations of the field. His work introduced key concepts such as schemes, sheaves, and cohomology, reshaping how mathematicians approach geometric problems and their algebraic underpinnings.
Algebraic closure: An algebraic closure of a field is an extension field in which every non-constant polynomial equation has a root. This concept ensures that any polynomial can be factored completely into linear factors, making the study of polynomial equations and their solutions more manageable. The algebraic closure is crucial in understanding the structure of fields, especially in the context of Galois theory and the behavior of morphisms in algebraic geometry.
Base Change: Base change is a process in algebraic geometry where one changes the base field or scheme over which a geometric object, such as a variety or a scheme, is defined. This operation allows mathematicians to study properties of the original object in relation to different contexts, revealing important insights into its behavior and structure, particularly when examining morphisms or relationships between varieties.
étale morphism: An étale morphism is a type of morphism between schemes that generalizes the notion of a locally isomorphic mapping in algebraic geometry. It is a flat morphism that is also unramified, meaning that it behaves like a local isomorphism in a way that avoids any 'branching' or singularities. This concept plays a crucial role in understanding properties of schemes, such as the behavior of points and the structure of fibers over different base schemes.
étale-locally criterion for separability: The étale-locally criterion for separability is a concept in algebraic geometry that provides a condition to determine whether a morphism between schemes is separable. Specifically, it states that a morphism is separable if, after passing to an étale cover, the corresponding ring homomorphism becomes a separable extension. This connection between étale morphisms and separability is crucial for understanding how algebraic structures behave over various fields and how they interact with each other.
Extension of scalars: An extension of scalars is a process in algebra where a mathematical structure, like a vector space or an algebra, is modified to allow coefficients from a larger field rather than just the original one. This method enables richer structures and interactions, especially in geometry and number theory, leading to various applications such as improving the study of Néron models and understanding étale morphisms.
Finite étale cover: A finite étale cover is a specific type of morphism in algebraic geometry that involves a finite number of maps from one scheme to another that are both flat and unramified. This means that the covering map behaves nicely, ensuring that locally on the target scheme, the preimage looks like a finite number of disjoint copies, which provides a strong notion of local triviality. Finite étale covers are crucial in understanding the structure of schemes and in lifting properties from the base to the cover.
Finiteness: Finiteness refers to the property of a mathematical object being limited or having a finite structure. In various areas of mathematics, this concept often implies that certain invariants or quantities associated with the object can be enumerated or counted, leading to useful results. Understanding finiteness helps mathematicians analyze and classify algebraic structures, geometric configurations, and morphisms effectively.
Flat morphism: A flat morphism is a type of morphism between schemes or algebraic varieties that preserves the dimension of fibers in a controlled way. It ensures that the fibers remain 'flat' over the base, meaning that they behave consistently when pulled back along any map. This property is important in various contexts, particularly when discussing étale morphisms, as it helps in understanding how structures interact in algebraic geometry.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his significant contributions to algebraic geometry, topology, and number theory. His work has deeply influenced various fields within mathematics, particularly in relation to the development of modern concepts and conjectures surrounding arithmetic geometry.
Local properties: Local properties refer to characteristics or behaviors of mathematical objects that can be observed by examining them in a neighborhood around a particular point. These properties are crucial because they often determine the behavior of objects in a broader context, such as how morphisms behave at certain points, which is especially relevant when discussing étale morphisms and their effects on schemes.
Scheme Morphism: A scheme morphism is a mathematical structure that describes a continuous and compatible way of mapping between two schemes, reflecting how their geometric and algebraic properties interact. This concept is foundational in algebraic geometry, as it allows for the exploration of relationships between different schemes, making it easier to study morphisms like étale morphisms, which maintain certain desirable properties such as flatness and smoothness.
Separation: In the context of étale morphisms, separation refers to a property of morphisms between schemes that ensures they behave nicely with respect to the underlying topological spaces. Specifically, a morphism is said to be separated if the diagonal morphism, which identifies points in the product space, is a closed immersion. This concept helps us understand how schemes can be distinguished from one another and ensures that various geometric properties are maintained in the transition between different schemes.
Spectrum of a ring: The spectrum of a ring, denoted as Spec(R), is the set of all prime ideals of a ring R, equipped with the Zariski topology. This construction allows for a geometric interpretation of algebraic structures, linking algebra to geometry by associating points in this space with prime ideals, thus providing insight into the structure and properties of the ring.
Unramified: In the context of algebraic geometry, unramified refers to a property of a morphism between schemes where the fibers above any point are discrete and the morphism behaves nicely with respect to differentials. This property implies that the morphism does not introduce any new infinitesimal elements, which means it can be viewed as 'locally' preserving the structure of the schemes involved. Unramified morphisms are significant in studying étale morphisms, as they relate to the local behavior and properties of algebraic varieties.
Zariski's Main Theorem: Zariski's Main Theorem establishes a powerful connection between the étale morphisms of schemes and the structure of their function fields. It states that under certain conditions, an étale morphism induces a bijection between the points of a scheme and the points of its base scheme, allowing for a comprehensive understanding of their geometric properties. This theorem is crucial for analyzing how different schemes relate to each other through étale morphisms.