Lemma 37.37.9. Let $S$ be a scheme. Let $Y \to X$ be a closed immersion of schemes smooth over $S$. For every $y \in Y$ there exist integers $0 \leq m, n$ and a commutative diagram

$\xymatrix{ Y \ar[d] & V \ar[l] \ar[d] \ar[r] & \mathbf{A}^ m_ S \ar[d]^{(a_1, \ldots , a_ m) \mapsto (a_1, \ldots , a_ m, 0 \ldots , 0)} \\ X & U \ar[l] \ar[r]^-\pi & \mathbf{A}^{m + n}_ S }$

where $U \subset X$ is open, $V = Y \cap U$, $\pi$ is étale, $V = \pi ^{-1}(\mathbf{A}^ m_ S)$, and $y \in V$.

Proof. The question is local on $X$ hence we may replace $X$ by an open neighbourhood of $y$. Since $Y \to X$ is a regular immersion by Divisors, Lemma 31.22.11 we may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine and there exists a regular sequence $f_1, \ldots , f_ n \in A$ such that $Y = V(f_1, \ldots , f_ n)$. After shrinking $X$ (and hence $Y$) further we may assume there exists an étale morphism $Y \to \mathbf{A}^ m_ S$, see Morphisms, Lemma 29.36.20. Let $\overline{g}_1, \ldots , \overline{g}_ m$ in $\mathcal{O}_ Y(Y)$ be the coordinate functions of this étale morphism. Choose lifts $g_1, \ldots , g_ m \in A$ of these functions and consider the morphism

$(g_1, \ldots , g_ m, f_1, \ldots , f_ n) : X \longrightarrow \mathbf{A}^{m + n}_ S$

over $S$. This is a morphism of schemes locally of finite presentation over $S$ and hence is locally of finite presentation (Morphisms, Lemma 29.21.11). The restriction of this morphism to $\mathbf{A}^ m_ S \subset \mathbf{A}^{m + n}_ S$ is étale by construction. Thus, in order to show that $X \to \mathbf{A}^{m + n}_ S$ is étale at $y$ it suffices to show that $X \to \mathbf{A}^{m + n}_ S$ is flat at $y$, see Morphisms, Lemma 29.36.15. Let $s \in S$ be the image of $y$. It suffices to check that $X_ s \to \mathbf{A}^{m + n}_ s$ is flat at $y$, see Theorem 37.16.2. Let $z \in \mathbf{A}^{m + n}_ s$ be the image of $y$. The local ring map

$\mathcal{O}_{\mathbf{A}^{m + n}_ s, z} \longrightarrow \mathcal{O}_{X_ s, y}$

is flat by Algebra, Lemma 10.128.1. Namely, schemes smooth over fields are regular and regular rings are Cohen-Macaulay, see Varieties, Lemma 33.25.3 and Algebra, Lemma 10.106.3. Thus both source and target are regular local rings (and hence CM). The source and target have the same dimension: namely, we have $\dim (\mathcal{O}_{Y_ s, y}) = \dim (\mathcal{O}_{\mathbf{A}^ m_ s, z})$ by More on Algebra, Lemma 15.44.2, we have $\dim (\mathcal{O}_{\mathbf{A}^{m + n}_ s, z}) = n + \dim (\mathcal{O}_{\mathbf{A}^ m_ s, z})$, and we have $\dim (\mathcal{O}_{X_ s, y}) = n + \dim (\mathcal{O}_{Y_ s, y})$ because $\mathcal{O}_{Y_ s, y}$ is the quotient of $\mathcal{O}_{X_ s, y}$ by the regular sequence $f_1, \ldots , f_ n$ of length $n$ (see Divisors, Remark 31.22.5). Finally, the fibre ring of the displayed arrow is finite over $\kappa (z)$ since $Y_ s \to \mathbf{A}^ m_ s$ is étale at $y$. This finishes the proof. $\square$

## 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 0FUE. Beware of the difference between the letter 'O' and the digit '0'.