The Stacks project

29.35 Étale morphisms

The Zariski topology of a scheme is a very coarse topology. This is particularly clear when looking at varieties over $\mathbf{C}$. It turns out that declaring an étale morphism to be the analogue of a local isomorphism in topology introduces a much finer topology. On varieties over $\mathbf{C}$ this topology gives rise to the “correct” Betti numbers when computing cohomology with finite coefficients. Another observable is that if $f : X \to Y$ is an étale morphism of varieties over $\mathbf{C}$, and if $x$ is a closed point of $X$, then $f$ induces an isomorphism $\mathcal{O}^{\wedge }_{Y, f(x)} \to \mathcal{O}^{\wedge }_{X, x}$ of complete local rings.

In this section we start our study of these matters. In fact we deliberately restrict our discussion to a minimum since we will discuss more interesting results elsewhere. Recall that a ring map $R \to A$ is said to be étale if it is smooth and $\Omega _{A/R} = 0$, see Algebra, Definition 10.142.1.

Definition 29.35.1. Let $f : X \to S$ be a morphism of schemes.

  1. We say that $f$ is étale at $x \in X$ if there exists an affine open neighbourhood $\mathop{\mathrm{Spec}}(A) = U \subset X$ of $x$ and affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is étale.

  2. We say that $f$ is étale if it is étale at every point of $X$.

  3. A morphism of affine schemes $f : X \to S$ is called standard étale if $X \to S$ is isomorphic to

    \[ \mathop{\mathrm{Spec}}(R[x]_ h/(g)) \to \mathop{\mathrm{Spec}}(R) \]

    where $R \to R[x]_ h/(g)$ is a standard étale ring map, see Algebra, Definition 10.143.1, i.e., $g$ is monic and $g'$ invertible in $R[x]_ h/(g)$.

A morphism is étale if and only if it is smooth of relative dimension $0$ (see Definition 29.33.13). A pleasing feature of the definition is that the set of points where a morphism is étale is automatically open.

Note that there is no separation or quasi-compactness hypotheses in the definition. Hence the question of being étale is local in nature on the source. Here is the precise result.

Lemma 29.35.2. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is étale.

  2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is étale.

  3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is étale.

  4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that the ring map $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i)$ is étale, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is étale then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is étale.

Proof. This follows from Lemma 29.14.3 if we show that the property “$R \to A$ is étale” is local. We check conditions (a), (b) and (c) of Definition 29.14.1. These all follow from Algebra, Lemma 10.142.3. $\square$

Lemma 29.35.3. The composition of two morphisms which are étale is étale.

Proof. In the proof of Lemma 29.35.2 we saw that being étale is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 29.14.5 combined with the fact that being étale is a property of ring maps that is stable under composition, see Algebra, Lemma 10.142.3. $\square$

Lemma 29.35.4. The base change of a morphism which is étale is étale.

Proof. In the proof of Lemma 29.35.2 we saw that being étale is a local property of ring maps. Hence the lemma follows from Lemma 29.14.5 combined with the fact that being étale is a property of ring maps that is stable under base change, see Algebra, Lemma 10.142.3. $\square$

Lemma 29.35.5. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Then $f$ is étale at $x$ if and only if $f$ is smooth and unramified at $x$.

Proof. This follows immediately from the definitions. $\square$

Proof. By Lemma 29.35.5 an étale morphism is unramified. By Lemma 29.34.10 an unramified morphism is locally quasi-finite. $\square$

slogan

Lemma 29.35.7. Fibres of étale morphisms.

  1. Let $X$ be a scheme over a field $k$. The structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is étale if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$.

  2. If $f : X \to S$ is an étale morphism, then for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$.

Proof. You can deduce this from Lemma 29.34.11 via Lemma 29.35.5 above. Here is a direct proof.

We will use Algebra, Lemma 10.142.4. Hence it is clear that if $X$ is a disjoint union of spectra of finite separable field extensions of $k$ then $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Conversely, suppose that $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Then for any affine open $U \subset X$ we see that $U$ is a finite disjoint union of spectra of finite separable field extensions of $k$. Hence all points of $X$ are closed points (see Lemma 29.20.2 for example). Thus $X$ is a discrete space and we win. $\square$

The following lemma characterizes an étale morphism as a flat, finitely presented morphism with “étale fibres”.

Lemma 29.35.8. Let $f : X \to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$, then $f$ is étale.

Proof. You can deduce this from Algebra, Lemma 10.142.7. Here is another proof.

By Lemma 29.35.7 a fibre $X_ s$ is étale and hence smooth over $s$. By Lemma 29.33.3 we see that $X \to S$ is smooth. By Lemma 29.34.12 we see that $f$ is unramified. We conclude by Lemma 29.35.5. $\square$

Proof. This is true because an open immersion is a local isomorphism. $\square$

Proof. See Algebra, Lemma 10.136.10 and use that an étale morphism is the same as a smooth morphism of relative dimension $0$. $\square$

Proof. True because an étale ring map is of finite presentation by definition. $\square$

The following lemma says locally any étale morphism is standard étale. This is actually kind of a tricky result to prove in complete generality. The tricky parts are hidden in the chapter on commutative algebra. Hence a standard étale morphism is a local model for a general étale morphism.

Lemma 29.35.14. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Let $V \subset S$ be an affine open neighbourhood of $f(x)$. The following are equivalent

  1. The morphism $f$ is étale at $x$.

  2. There exist an affine open $U \subset X$ with $x \in U$ and $f(U) \subset V$ such that the induced morphism $f|_ U : U \to V$ is standard étale (see Definition 29.35.1).

Proof. Follows from the definitions and Algebra, Proposition 10.143.4. $\square$

Here is a differential criterion of étaleness at a point. There are many variants of this result all of which may be useful at some point. We will just add them here as needed.

Lemma 29.35.15. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Set $s = f(x)$. Assume $f$ is locally of finite presentation. The following are equivalent:

  1. The morphism $f$ is étale at $x$.

  2. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $X_ s \to \mathop{\mathrm{Spec}}(\kappa (s))$ is étale at $x$.

  3. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and $X_ s \to \mathop{\mathrm{Spec}}(\kappa (s))$ is unramified at $x$.

  4. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\mathcal{O}_{X, x}$-module $\Omega _{X/S, x}$ is zero.

  5. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat and the $\kappa (x)$-vector space

    \[ \Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) = \Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x) \]

    is zero.

  6. The local ring map $\mathcal{O}_{S, s} \to \mathcal{O}_{X, x}$ is flat, we have $\mathfrak m_ s\mathcal{O}_{X, x} = \mathfrak m_ x$ and the field extension $\kappa (s) \subset \kappa (x)$ is finite separable.

  7. There exist affine opens $U \subset X$, and $V \subset S$ such that $x \in U$, $f(U) \subset V$ and the induced morphism $f|_ U : U \to V$ is standard smooth of relative dimension $0$.

  8. There exist affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$ and $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $x \in U$ corresponding to $\mathfrak q \subset A$, and $f(U) \subset V$ such that there exists a presentation

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

    with

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

    mapping to an element of $A$ not in $\mathfrak q$.

  9. There exist affine opens $U \subset X$, and $V \subset S$ such that $x \in U$, $f(U) \subset V$ and the induced morphism $f|_ U : U \to V$ is standard étale.

  10. There exist affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$ and $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $x \in U$ corresponding to $\mathfrak q \subset A$, and $f(U) \subset V$ such that there exists a presentation

    \[ A = R[x]_ Q/(P) = R[x, 1/Q]/(P) \]

    with $P, Q \in R[x]$, $P$ monic and $P' = \text{d}P/\text{d}x$ mapping to an element of $A$ not in $\mathfrak q$.

Proof. Use Lemma 29.35.14 and the definitions to see that (1) implies all of the other conditions. For each of the conditions (2) – (10) combine Lemmas 29.33.14 and 29.34.14 to see that (1) holds by showing $f$ is both smooth and unramified at $x$ and applying Lemma 29.35.5. Some details omitted. $\square$

Lemma 29.35.16. A morphism is étale at a point if and only if it is flat and G-unramified at that point. A morphism is étale if and only if it is flat and G-unramified.

Lemma 29.35.17. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]

be a cartesian diagram of schemes. Let $W \subset X$, resp. $W' \subset X'$ be the open subscheme of points where $f$, resp. $f'$ is étale. Then $W' = (g')^{-1}(W)$ if

  1. $f$ is flat and locally of finite presentation, or

  2. $f$ is locally of finite presentation and $g$ is flat.

Proof. Assume first that $f$ locally of finite type. Consider the set

\[ T = \{ x \in X \mid f\text{ is unramified at }x\} \]

and the corresponding set $T' \subset X'$ for $f'$. Then $T' = (g')^{-1}(T)$ by Lemma 29.34.15.

Thus case (1) follows because in case (1) $T$ is the (open) set of points where $f$ is étale by Lemma 29.35.16.

In case (2) let $x' \in W'$. Then $g'$ is flat at $x'$ (Lemma 29.25.7) and $g \circ f'$ is flat at $x'$ (Lemma 29.25.5). It follows that $f$ is flat at $x = g'(x')$ by Lemma 29.25.13. On the other hand, since $x' \in T'$ (Lemma 29.33.5) we see that $x \in T$. Hence $f$ is étale at $x$ by Lemma 29.35.15. $\square$

Our proof of the following lemma is somewhat complicated. It uses the “Critère de platitude par fibres” to see that a morphism $X \to Y$ over $S$ between schemes étale over $S$ is automatically flat. The details are in the chapter on commutative algebra.

slogan

Lemma 29.35.18. Let $f : X \to Y$ be a morphism of schemes over $S$. If $X$ and $Y$ are étale over $S$, then $f$ is étale.

Proof. See Algebra, Lemma 10.142.8. $\square$

Lemma 29.35.19. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

  1. $f$ is surjective, and étale,

  2. $p$ is étale, and

  3. $q$ is locally of finite presentation1.

Then $q$ is étale.

Proof. By Lemma 29.33.19 we see that $q$ is smooth. Thus we only need to see that $q$ has relative dimension $0$. This follows from Lemma 29.28.2 and the fact that $f$ and $p$ have relative dimension $0$. $\square$

A final characterization of smooth morphisms is that a smooth morphism $f : X \to S$ is locally the composition of an étale morphism by a projection $\mathbf{A}_ S^ d \to S$.

slogan

Lemma 29.35.20. Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. Let $V \subset Y$ be an affine open neighbourhood of $f(x)$. If $\varphi $ is smooth at $x$, then there exists an integer $d \geq 0$ and an affine open $U \subset X$ with $x \in U$ and $\varphi (U) \subset V$ such that there exists a commutative diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \ar[r]_-\pi & \mathbf{A}^ d_ V \ar[ld] \\ Y & V \ar[l] } \]

where $\pi $ is étale.

Proof. By Lemma 29.33.11 we can find an affine open $U$ as in the lemma such that $\varphi |_ U : U \to V$ is standard smooth. Write $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(R)$ so that we can write

\[ A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c) \]

with

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

mapping to an invertible element of $A$. Then it is clear that $R[x_{c + 1}, \ldots , x_ n] \to A$ is standard smooth of relative dimension $0$. Hence it is smooth of relative dimension $0$. In other words the ring map $R[x_{c + 1}, \ldots , x_ n] \to A$ is étale. As $\mathbf{A}^{n - c}_ V = \mathop{\mathrm{Spec}}(R[x_{c + 1}, \ldots , x_ n])$ the lemma with $d = n - c$. $\square$

[1] In fact this is implied by (1) and (2), see Descent, Lemma 35.11.3. Moreover, it suffices to assume that $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma 35.11.5.

Comments (3)

Comment #5503 by Thor Wittich on

I might be wrong but I think condition (8) in 02GU does only characterize smoothness at the point x. There should also be something like "relative dimension zero" there.

Comment #5507 by Manuel Hoff on

@Thor Wittich: The characterisation is correct like this. The "relative dimension 0"-part is encoded by the matrix being a square matrix (i.e. the number of variables is equal to the number of polynomials that are quotiened out).

Comment #5515 by Thor Wittich on

@Manuel Hoff You are totally right. I was just blind. Thanks!


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