The Stacks project

29.36 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 10.143.1.

Definition 29.36.1. 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 10.144.1, 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 29.34.13). 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 29.36.2. 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 10.143.3. $\square$

Lemma 29.36.3. The composition of two morphisms which are étale is étale.

Proof. In the proof of Lemma 29.36.2 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 10.143.3. $\square$

Lemma 29.36.4. The base change of a morphism which is étale is étale.

Proof. In the proof of Lemma 29.36.2 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 10.143.3. $\square$

Lemma 29.36.5. 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 29.36.5 an étale morphism is unramified. By Lemma 29.35.10 an unramified morphism is locally quasi-finite. $\square$

slogan

Lemma 29.36.7. 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 29.35.11 via Lemma 29.36.5 above. Here is a direct proof.

We will use Algebra, Lemma 10.143.4. 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 29.36.8. 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 10.143.7. Here is another proof.

By Lemma 29.36.7 a fibre $X_ s$ is étale and hence smooth over $s$. By Lemma 29.34.3 we see that $X \to S$ is smooth. By Lemma 29.35.12 we see that $f$ is unramified. We conclude by Lemma 29.36.5. $\square$

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

Proof. See Algebra, Lemma 10.137.10 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 29.36.14. 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 29.36.1).

Proof. Follows from the definitions and Algebra, Proposition 10.144.4. $\square$

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 29.36.15. 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 $\kappa (x)/\kappa (s)$ 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 29.36.14 and the definitions to see that (1) implies all of the other conditions. For each of the conditions (2) – (10) combine Lemmas 29.34.14 and 29.35.14 to see that (1) holds by showing $f$ is both smooth and unramified at $x$ and applying Lemma 29.36.5. Some details omitted. $\square$

Lemma 29.36.16. 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 29.36.17. 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 29.35.15.

Thus case (1) follows because in case (1) $T$ is the (open) set of points where $f$ is étale by Lemma 29.36.16.

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 29.34.5) we see that $x \in T$. Hence $f$ is étale at $x$ by Lemma 29.36.15. $\square$

slogan

Lemma 29.36.18. 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. As a first proof one may reduce to the affine case and then use Algebra, Lemma 10.143.8. This proof is somewhat complicated as 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.

We give a second proof using the graph argument. Namely, consider the factorization $X \to X \times _ S Y \to Y$, where the first arrow is given by $\text{id}_ X$ and $f$ and the second arrow is the projection. We claim both arrows are étale and hence $f$ is étale by Lemma 29.36.3. Namely, the projection is étale as it is the base change of $X \to S$, see Lemma 29.36.4. The first arrow is the base change of the diagonal morphism $Y \to Y \times _ S Y$ because the square

\[ \xymatrix{ X \ar[d] \ar[r] & X \times _ S Y \ar[d] \\ Y \ar[r] & Y \times _ S Y } \]

is cartesian. The diagonal $Y \to Y \times _ S Y$ is an open immersion because $Y \to S$ is étale and hence unramified (Lemma 29.36.5) and we may use Lemma 29.35.13. The base change of an open immersion is an open immersion (Schemes, Lemma 26.18.2) and an open immersion is étale (Lemma 29.36.9). This finishes the second proof. $\square$

Lemma 29.36.19. 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 29.34.19 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 29.36.20. Let $\varphi : X \to Y$ be a morphism of schemes. Let $x \in X$. Let $V \subset Y$ be an affine open neighbourhood of $\varphi (x)$. 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 29.34.11 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 35.14.3. Moreover, it suffices to assume that $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma 35.14.5.

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!

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02GH. Beware of the difference between the letter 'O' and the digit '0'.



View Section 29.36 as pdf