Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Étale morphism
Summary
In algebraic geometry, an étale morphism (etal) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not necessarily local isomorphisms. Despite this, étale maps retain many of the properties of local analytic isomorphisms, and are useful in defining the algebraic fundamental group and the étale topology.
The word étale is a French adjective, which means "slack", as in "slack tide", or, figuratively, calm, immobile, something left to settle.
Let be a ring homomorphism. This makes an -algebra. Choose a monic polynomial in and a polynomial in such that the derivative of is a unit in . We say that is standard étale if and can be chosen so that is isomorphic as an -algebra to and is the canonical map.
Let be a morphism of schemes. We say that is étale if and only if it has any of the following equivalent properties:
is flat and unramified.
is a smooth morphism and unramified.
is flat, locally of finite presentation, and for every in , the fiber is the disjoint union of points, each of which is the spectrum of a finite separable field extension of the residue field .
is flat, locally of finite presentation, and for every in and every algebraic closure of the residue field , the geometric fiber is the disjoint union of points, each of which is isomorphic to .
is a smooth morphism of relative dimension zero.
is a smooth morphism and a locally quasi-finite morphism.
is locally of finite presentation and is locally a standard étale morphism, that is,
For every in , let . Then there is an open affine neighborhood Spec R of and an open affine neighborhood Spec S of such that f(Spec S) is contained in Spec R and such that the ring homomorphism R → S induced by is standard étale.
is locally of finite presentation and is formally étale.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood
Related courses (10)
MATH-436: Homotopical algebra
This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous
MATH-110(a): Advanced linear algebra I
L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et de démontrer rigoureusement les résultats principaux de ce sujet.
MATH-211: Group Theory
Après une introduction à la théorie des catégories, nous appliquerons la théorie générale au cas particulier des groupes, ce qui nous permettra de bien mettre en perspective des notions telles que quo
Related lectures (68)
Groups & Subgroups: Definitions, Theorems, and Morphisms
Covers the fundamental concepts of groups and subgroups, including definitions, theorems, and morphisms.
Finite Maps: Morphism of Schemes
Covers morphism of schemes, affine covering, integral homomorphism, and properties of finite maps.
Geometric Objects in Algebra
Covers the study of graded modules and quasicoherent sheaves in algebraic geometry.
Related publications (21)
Co-Design Optimisation of Morphing Topology and Control of Winged Drones
Dario Floreano, Valentin Wüest, Fabio Bergonti
The design and control of winged aircraft and drones is an iterative process aimed at identifying a compromise of mission-specific costs and constraints. When agility is required, shape-shifting (morphing) drones represent an efficient solution. However, m ...
2024Cyclic $A_\infty$-algebras and cyclic homology
We provide a new description of the complex computing the Hochschild homology of an -unitary -algebra as a derived tensor product such that: (1) there is a canonical morphism from it to the complex computing the cyclic homology of that was introduced by Ko ...
2023NORM TORI OF ETALE ALGEBRAS AND UNRAMIFIED BRAUER GROUPS
Eva Bayer Fluckiger, Ting-Yu Lee
Let k be a field, and let L be an etale k-algebra of finite rank. If a is an element of k(x), let X-a be the affine variety defined by N-L/k(x) = a. Assuming that L has at least one factor that is a cyclic field extension of k, we give a combinatorial desc ...
Jerusalem2023Related people (2)
Related concepts (16)
Glossary of algebraic geometry
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.
Algebraic space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology.
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).