Lemma 37.17.1. 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
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