# Étale morphism

An **étale morphism** is a morphism of schemes that is smooth of relative dimension 0, the algebraic-geometric analog of a local diffeomorphism or a covering map in topology.<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> The word is French for "spread out" or "slack"; Grothendieck chose the adjective in analogy with the classical notion of a *domaine étalé* in complex analytic geometry.<sup>[2](https://www.imo.universite-paris-saclay.fr/~luc.illusie/Grothendieck_etale.pdf)</sup> Étale morphisms are the building blocks of the étale topology and étale cohomology, the framework in which Grothendieck and collaborators developed a cohomology theory for schemes in the Séminaire de Géométrie Algébrique (SGA), beginning with SGA 1 in 1960–61.<sup>[3](https://arxiv.org/abs/math/0206203)</sup>

| Key fact | Detail |
|---|---|
| Definition | A morphism of schemes is étale if it is smooth of relative dimension 0; equivalently, smooth with \( \Omega_{X/S} = 0 \)<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> |
| Equivalent forms | Locally finitely presented, flat, and all fibers étale; or flat, unramified, and locally of finite presentation<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> |
| Over a field | \( U \to \mathrm{Spec}(k) \) is étale exactly when \( U \cong \coprod_i \mathrm{Spec}(k_i) \) with each \( k_i \) a finite separable extension of \( k \)<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> |
| Jacobian criterion | \( A \to B \) is étale iff \( B \cong A[x_1,\ldots,x_n]/(f_1,\ldots,f_n) \) with \( \det(\partial f_i/\partial x_j) \) invertible in \( B \)<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> |
| Stability | Base changes, compositions, and fiber products of étale morphisms are étale<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> |
| Lifting property | For finite presentation ring maps, étale is equivalent to formally étale, the unique lifting of maps through square-zero ideals<sup>[4](https://stacks.math.columbia.edu/tag/00UP)</sup> |
| Role in the étale site | Étale covers, jointly surjective families of étale morphisms locally of finite type, define the étale topology<sup>[5](https://ncatlab.org/nlab/show/%C3%A9tale%2Bcover)</sup> |

## Definition and equivalent characterizations

The Stacks project defines smoothness first: a ring map \( A \to B \) is smooth if it is flat, finitely presented, and every fiber ring \( \kappa(\mathfrak p) \otimes_A B \) is smooth over the residue field; a morphism of schemes is smooth if it is flat, locally of finite presentation, and the geometric fibers are smooth. An étale morphism is then a smooth morphism of relative dimension 0, and in particular \( X \to S \) is étale if it is smooth and the module of relative differentials satisfies \( \Omega_{X/S} = 0 \).<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup>

Several independent characterizations agree. Proposition 59.26.2 of the Stacks project lists equivalent conditions on a morphism \( \varphi : U \to S \): it is étale; it is locally finitely presented, flat, and all its fibers are étale; or it is flat, unramified, and locally of finite presentation.<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> The Stacks project's chapter on étale morphisms presents four ways of thinking about them: as smooth morphisms of relative dimension 0, as locally finitely presented flat unramified morphisms, through the structure theorem, and through the functorial characterization.<sup>[6](https://stacks.math.columbia.edu/tag/025J)</sup>

The **Jacobian criterion** gives a concrete computational test. A ring map \( A \to B \) is étale if and only if \( B \cong A[x_1, \ldots, x_n]/(f_1, \ldots, f_n) \) such that the determinant \( \Delta = \det\left( \frac{\partial f_i}{\partial x_j} \right) \) is invertible in \( B \).<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> For morphisms of affine varieties over a nonsingular base, the same condition on the invertibility of the Jacobian determinant characterizes étaleness.<sup>[7](https://ncatlab.org/nlab/show/etale%2Bmorphism%2Bof%2Bschemes)</sup>

## The case over a field

Over a field the notion reduces to separability. A morphism \( U \to \mathrm{Spec}(k) \) is étale if and only if \( U \) is a disjoint union \( \coprod_{i \in I} \mathrm{Spec}(k_i) \) where each \( k_i \) is a field that is a finite separable extension of \( k \).<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup>

## Formal étaleness and the functorial criterion

The **formally étale** property is defined by a lifting condition. A ring map \( R \to S \) is formally étale if, for every commutative diagram in which \( I \subset A \) is an ideal of square zero, there exists a unique dotted arrow \( S \to A \) making the diagram commute; equivalently, the map is both formally smooth and formally unramified.<sup>[4](https://stacks.math.columbia.edu/tag/00UP)</sup> For ring maps of finite presentation, formally étale and étale are equivalent: formal étaleness gives formal smoothness and \( \Omega_{S/R} = 0 \), which is étaleness by definition, and conversely an étale map is formally smooth and formally unramified.<sup>[4](https://stacks.math.columbia.edu/tag/00UP)</sup>

The corresponding **functorial characterization** for schemes states that a morphism \( f : X \to S \) locally of finite presentation is étale if and only if, for every affine \( S \)-scheme \( Y \) and every closed subscheme \( Y_0 \subset Y \) defined by a square-zero ideal, the natural map \( \mathrm{Mor}_S(Y, X) \to \mathrm{Mor}_S(Y_0, X) \) is bijective.<sup>[6](https://stacks.math.columbia.edu/tag/025J)</sup> In words, solutions to the equations defining \( X \) lift uniquely through nilpotent thickenings. In practice this criterion matters when showing an algebra is representable, since the corepresentable functor may be more accessible than the algebra itself.<sup>[8](https://math.uchicago.edu/~amathew/CRing.pdf)</sup>

## Stability properties

Étaleness is preserved by the standard operations of scheme theory: the base change of an étale morphism is étale, compositions of étale morphisms are étale, and fiber products and products of étale morphisms are étale.<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> An étale morphism always has relative dimension 0.<sup>[1](https://stacks.math.columbia.edu/tag/03PA)</sup> Formally étale maps enjoy the same base-change stability.<sup>[4](https://stacks.math.columbia.edu/tag/00UP)</sup>

## Role in the étale topology and cohomology

The étale topology on a scheme takes as its covering families the **étale covers**: sets of morphisms \( \{p_i : U_i \to X\} \) that are étale and locally of finite type and whose images jointly cover \( X \).<sup>[5](https://ncatlab.org/nlab/show/%C3%A9tale%2Bcover)</sup> Sheaves for this topology support étale cohomology, which behaves for many purposes like the cohomology of topological spaces; for example, a Mayer-Vietoris long exact sequence holds for a union of two opens.<sup>[9](https://stacks.math.columbia.edu/tag/03Q3)</sup> The Stacks project's chapter on étale morphisms states as its goal to collect enough commutative algebra to let a reader start on a treatise on étale cohomology.<sup>[10](https://stacks.math.columbia.edu/download/etale.pdf)</sup>

The theory extends beyond schemes. Étale morphisms of algebraic spaces are defined so as to agree with the existing notion for morphisms representable by algebraic spaces, using the fact that a pullback \( U \to X \) of a smooth surjective cover is étale.<sup>[11](https://stacks.math.columbia.edu/tag/0CIK)</sup> For algebraic stacks the naive definition fails: the Stacks project notes that an étale morphism of algebraic stacks should *not* be defined as a smooth morphism of relative dimension 0, because \( [\mathbf{A}^1_k/\mathbf{G}_{m,k}] \to \mathrm{Spec}(k) \) is smooth of relative dimension 0 yet does not identify tangent spaces, and in characteristic \( p > 0 \) the representable morphism \( \mathrm{Spec}(k) \to [\mathrm{Spec}(k)/\mu_{p,k}] \) is not étale since its base change \( \mu_{p,k} \to \mathrm{Spec}(k) \) comes from a nonreduced scheme. The definition instead requires étaleness to be preserved under base change and étale morphisms from stacks to schemes to be Deligne-Mumford.<sup>[11](https://stacks.math.columbia.edu/tag/0CIK)</sup>

## Intuition and comparison with other morphism classes

In number theory, the unramified locus of a morphism \( \mathrm{Spec}(\mathcal{O}_L) \to \mathrm{Spec}(\mathcal{O}_K) \) between rings of integers of number fields matches the conventional ramification theory of the field extension \( L/K \).<sup>[12](https://github.com/stacks/stacks-project/blob/master/etale.tex)</sup>

## References

1. [Section 59.26 (03PA): Étale morphisms, The Stacks project](https://stacks.math.columbia.edu/tag/03PA)
2. [Luc Illusie, Grothendieck et la cohomologie étale (on the choice of the adjective "étale")](https://www.imo.universite-paris-saclay.fr/~luc.illusie/Grothendieck_etale.pdf)
3. [Revêtements étales et groupe fondamental (SGA 1), arXiv:math/0206203](https://arxiv.org/abs/math/0206203)
4. [Section 10.150 (00UP): Formally étale maps, The Stacks project](https://stacks.math.columbia.edu/tag/00UP)
5. [étale cover in nLab](https://ncatlab.org/nlab/show/%C3%A9tale%2Bcover)
6. [Section 41.16 (025J): The functorial characterization, The Stacks project](https://stacks.math.columbia.edu/tag/025J)
7. [etale morphism of schemes in nlab](https://ncatlab.org/nlab/show/etale%2Bmorphism%2Bof%2Bschemes)
8. [Étale, unramified, and smooth morphisms, A. Mathew, University of Chicago](https://math.uchicago.edu/~amathew/CRing.pdf)
9. [Section 59.50 (03Q3): Étale cohomology, The Stacks project](https://stacks.math.columbia.edu/tag/03Q3)
10. [ÉTALE MORPHISMS OF SCHEMES, chapter PDF, The Stacks project](https://stacks.math.columbia.edu/download/etale.pdf)
11. [Section 101.35 (0CIK): Étale morphisms of algebraic stacks, The Stacks project](https://stacks.math.columbia.edu/tag/0CIK)
12. [stacks-project/etale.tex, GitHub (number-theoretic example of unramified morphisms)](https://github.com/stacks/stacks-project/blob/master/etale.tex)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Algebraic geometry › Schemes, stacks, and morphisms › Properties of morphisms*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license