The Stacks project

Lemma 37.38.6. Let $f : X \to S$ be a smooth morphism of schemes. Let $s \in S$ be a point in the image of $f$. Then there exists an étale neighbourhood $(S', s') \to (S, s)$ and a $S$-morphism $S' \to X$.

First proof of Lemma 37.38.6. By assumption $X_ s \not= \emptyset $. By Varieties, Lemma 33.25.6 there exists a closed point $x \in X_ s$ such that $\kappa (x)$ is a finite separable field extension of $\kappa (s)$. Hence by Lemma 37.38.5 there exists an immersion $Z \to X$ such that $Z \to S$ is étale and such that $x \in Z$. Take $(S' , s') = (Z, x)$. $\square$

Second proof of Lemma 37.38.6. Pick a point $x \in X$ with $f(x) = s$. Choose a diagram

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

with $\pi $ étale, $x \in U$ and $V = \mathop{\mathrm{Spec}}(R)$ affine, see Morphisms, Lemma 29.36.20. In particular $s \in V$. The morphism $\pi : U \to \mathbf{A}^ d_ V$ is open, see Morphisms, Lemma 29.36.13. Thus $W = \pi (U) \cap \mathbf{A}^ d_ s$ is a nonempty open subset of $\mathbf{A}^ d_ s$. Let $w \in W$ be a point with $\kappa (s) \subset \kappa (w)$ finite separable, see Varieties, Lemma 33.25.5. By Algebra, Lemma 10.114.1 there exist $d$ elements $\overline{f}_1, \ldots , \overline{f}_ d \in \kappa (s)[x_1, \ldots , x_ d]$ which generate the maximal ideal corresponding to $w$ in $\kappa (s)[x_1, \ldots , x_ d]$. After replacing $R$ by a principal localization we may assume there are $f_1, \ldots , f_ d \in R[x_1, \ldots , x_ d]$ which map to $\overline{f}_1, \ldots , \overline{f}_ d \in \kappa (s)[x_1, \ldots , x_ d]$. Consider the $R$-algebra

\[ R' = R[x_1, \ldots , x_ d]/(f_1, \ldots , f_ d) \]

and set $S' = \mathop{\mathrm{Spec}}(R')$. By construction we have a closed immersion $j : S' \to \mathbf{A}^ d_ V$ over $V$. By construction the fibre of $S' \to V$ over $s$ is a single point $s'$ whose residue field is finite separable over $\kappa (s)$. Let $\mathfrak q' \subset R'$ be the corresponding prime. By Algebra, Lemma 10.136.10 we see that $(R')_ g$ is a relative global complete intersection over $R$ for some $g \in R'$, $g \not\in \mathfrak q$. Thus $S' \to V$ is flat and of finite presentation in a neighbourhood of $s'$, see Algebra, Lemma 10.136.13. By construction the scheme theoretic fibre of $S' \to V$ over $s$ is $\mathop{\mathrm{Spec}}(\kappa (s'))$. Hence it follows from Morphisms, Lemma 29.36.15 that $S' \to S$ is étale at $s'$. Set

\[ S'' = U \times _{\pi , \mathbf{A}^ d_ V, j} S'. \]

By construction there exists a point $s'' \in S''$ which maps to $s'$ via the projection $p : S'' \to S'$. Note that $p$ is étale as the base change of the étale morphism $\pi $, see Morphisms, Lemma 29.36.4. Choose a small affine neighbourhood $S''' \subset S''$ of $s''$ which maps into the nonempty open neighbourhood of $s' \in S'$ where the morphism $S' \to S$ is étale. Then the étale neighbourhood $(S''', s'') \to (S, s)$ is a solution to the problem posed by the lemma. $\square$

Comments (6)

Comment #3822 by Johannes Anschuetz on

The "Y" in the diagram seems to mean "S".

Comment #4927 by robot0079 on

Here is a quick proof.

We can assume is surjective.

When S is a strict henselian local ring, this result follows from the fact that is surjective, where k is residue field of S. We can prove this by factoring it into composition of etale and relative affine space morphism.

Now in general case. From above we see that is non empty, we deduce our result by limit preserving property of morphism of locally of finite presentation.

Comment #5195 by on

@#4927. Yes, this is an outline of a proof.

Comment #5970 by Dario Weißmann on

typo: W should be defined as instead of

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