The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 36.34.6. 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 36.34.6. By assumption $X_ s \not= \emptyset $. By Varieties, Lemma 32.25.6 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 36.34.5 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 36.34.6. Pick a point $x \in X$ with $f(x) = s$. Choose a diagram

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

with $\pi $ étale, $x \in U$ and $V = \mathop{\mathrm{Spec}}(R)$ affine, see Morphisms, Lemma 28.34.20. In particular $s \in V$. The morphism $\pi : U \to \mathbf{A}^ d_ V$ is open, see Morphisms, Lemma 28.34.13. Thus $W = \pi (V) \cap \mathbf{A}^ d_ s$ 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 32.25.5. By Algebra, Lemma 10.113.1 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 10.134.11 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 10.134.14. By construction the scheme theoretic fibre of $S' \to V$ over $s$ is $\mathop{\mathrm{Spec}}(\kappa (s'))$. Hence it follows from Morphisms, Lemma 28.34.15 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 28.34.4. 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 (1)

Comment #3822 by Johannes Anschuetz on

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


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