## 41.11 Étale morphisms

In this section, we will define étale morphisms and prove a number of important properties about them. The most important one, no doubt, is the functorial characterization presented in Theorem 41.16.1. Following this, we will also discuss a few properties of rings which are insensitive to an étale extension (properties which hold for a ring if and only if they hold for all its étale extensions) to motivate the basic tenet of étale cohomology – étale morphisms are the algebraic analogue of local isomorphisms.

As the title suggests, we will define the class of étale morphisms – the class of morphisms (whose surjective families) we shall deem to be coverings in the category of schemes over a base scheme $S$ in order to define the étale site $S_{\acute{e}tale}$. Intuitively, an étale morphism is supposed to capture the idea of a covering space and, therefore, should be close to a local isomorphism. If we're working with varieties over algebraically closed fields, this last statement can be made into a definition provided we replace “local isomorphism” with “formal local isomorphism” (isomorphism after completion). One can then give a definition over any base field by asking that the base change to the algebraic closure be étale (in the aforementioned sense). But, rather than proceeding via such aesthetically displeasing constructions, we will adopt a cleaner, albeit slightly more abstract, algebraic approach.

We first define “étale homomorphisms of local rings” for Noetherian local rings. We cannot use the term “étale”, as there already is a notion of an étale ring map (Algebra, Section 10.143) and it is different.

Definition 41.11.1. Let $A$, $B$ be Noetherian local rings. A local homomorphism $f : A \to B$ is said to be an étale homomorphism of local rings if it is flat and an unramified homomorphism of local rings (please see Definition 41.3.1).

This is the local version of the definition of an étale ring map in Algebra, Section 10.143. The exact definition given in that section is that it is a smooth ring map of relative dimension $0$. It is shown (in Algebra, Lemma 10.143.2) that an étale $R$-algebra $S$ always has a presentation

$S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)$

such that

$g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_ n/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_ n/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_ n & \partial f_2/\partial x_ n & \ldots & \partial f_ n/\partial x_ n \end{matrix} \right)$

maps to an invertible element in $S$. The following two lemmas link the two notions.

Lemma 41.11.2. Let $A \to B$ be of finite type with $A$ a Noetherian ring. Let $\mathfrak q$ be a prime of $B$ lying over $\mathfrak p \subset A$. Then $A \to B$ is étale at $\mathfrak q$ if and only if $A_{\mathfrak p} \to B_{\mathfrak q}$ is an étale homomorphism of local rings.

Proof. See Algebra, Lemmas 10.143.3 (flatness of étale maps), 10.143.5 (étale maps are unramified) and 10.143.7 (flat and unramified maps are étale). $\square$

Lemma 41.11.3. Let $A$, $B$ be Noetherian local rings. Let $A \to B$ be a local homomorphism such that $B$ is essentially of finite type over $A$. The following are equivalent

1. $A \to B$ is an étale homomorphism of local rings

2. $A^\wedge \to B^\wedge$ is an étale homomorphism of local rings, and

3. $A^\wedge \to B^\wedge$ is étale.

Moreover, in this case $B^\wedge \cong (A^\wedge )^{\oplus n}$ as $A^\wedge$-modules for some $n \geq 1$.

Proof. To see the equivalences of (1), (2) and (3), as we have the corresponding results for unramified ring maps (Lemma 41.3.4) it suffices to prove that $A \to B$ is flat if and only if $A^\wedge \to B^\wedge$ is flat. This is clear from our lists of properties of flat maps since the ring maps $A \to A^\wedge$ and $B \to B^\wedge$ are faithfully flat. For the final statement, by Lemma 41.3.3 we see that $B^\wedge$ is a finite flat $A^\wedge$ module. Hence it is finite free by our list of properties on flat modules in Section 41.9. $\square$

The integer $n$ which occurs in the lemma above is nothing other than the degree $[\kappa (\mathfrak m_ B) : \kappa (\mathfrak m_ A)]$ of the residue field extension. In particular, if $\kappa (\mathfrak m_ A)$ is separably closed, we see that $A^\wedge \to B^\wedge$ is an isomorphism, which vindicates our earlier claims.

Definition 41.11.4. (See Morphisms, Definition 29.36.1.) Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism of schemes which is locally of finite type.

1. Let $x \in X$. We say $f$ is étale at $x \in X$ if $\mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X, x}$ is an étale homomorphism of local rings.

2. The morphism is said to be étale if it is étale at all its points.

Let us prove that this definition agrees with the definition in the chapter on morphisms of schemes. This in particular guarantees that the set of points where a morphism is étale is open.

Lemma 41.11.5. Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be locally of finite type. Let $x \in X$. The morphism $f$ is étale at $x$ in the sense of Definition 41.11.4 if and only if it is étale at $x$ in the sense of Morphisms, Definition 29.36.1.

Proof. This follows from Lemma 41.11.2 and the definitions. $\square$

Here are some results on étale morphisms. The formulations as given in this list apply only to morphisms locally of finite type between locally Noetherian schemes. In each case we give a reference to the general result as proved earlier in the project, but in some cases one can prove the result more easily in the Noetherian case. Here is the list:

1. An étale morphism is unramified. (Clear from our definitions.)

2. Étaleness is local on the source and the target in the Zariski topology.

3. Étale morphisms are stable under base change and composition. See Morphisms, Lemmas 29.36.4 and 29.36.3.

4. Étale morphisms of schemes are locally quasi-finite and quasi-compact étale morphisms are quasi-finite. (This is true because it holds for unramified morphisms as seen earlier.)

5. Étale morphisms have relative dimension $0$. See Morphisms, Definition 29.29.1 and Morphisms, Lemma 29.29.5.

6. A morphism is étale if and only if it is flat and all its fibres are étale. See Morphisms, Lemma 29.36.8.

7. Étale morphisms are open. This is true because an étale morphism is flat, and Theorem 41.10.2.

8. Let $X$ and $Y$ be étale over a base scheme $S$. Any $S$-morphism from $X$ to $Y$ is étale. See Morphisms, Lemma 29.36.18.

Comment #1934 by Dragos on

I think "flat at x" is missing from the statement of Lemma 40.11.5.

Comment #1993 by on

Dear Dragos, yes that would work, but actually we want to compare the two definitions so it should say etale and not unramified. See change here.

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