37.17 Closed immersions between smooth schemes

Some results that do not fit elsewhere very well.

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. $\square$

Remark 37.17.2. We fix a ring $R$ and we set $S = \mathop{\mathrm{Spec}}(R)$. Fix integers $0 \leq m$ and $1 \leq n$. Consider the closed immersion

$Z = \mathbf{A}^ m_ S \longrightarrow \mathbf{A}^{m + n}_ S = X,\quad (a_1, \ldots , a_ m) \mapsto (a_1, \ldots , a_ m, 0, \ldots 0).$

We are going to consider the blowing up $X'$ of $X$ along the closed subscheme $Z$. Write

$X = \mathop{\mathrm{Spec}}(A) \quad \text{with}\quad A = R[x_1, \ldots , x_ m, y_1, \ldots , y_ n]$

Then $X'$ is the Proj of the Rees algebra of $A$ with respect ot the ideal $(y_1, \ldots , y_ n)$. This Rees algebra is equal to $B = A[T_1, \ldots , T_ n]/(y_ iT_ j - y_ jT_ i)$; details omitted. Hence $X' = \text{Proj}(B)$ is smooth over $S$ as it is covered by the affine opens

\begin{align*} D_+(T_ i) & = \mathop{\mathrm{Spec}}(B_{(T_ i)}) \\ & = \mathop{\mathrm{Spec}}(A[t_1, \ldots , \hat t_ i, \ldots t_ n]/(y_ j - y_ i t_ j)) \\ & = \mathop{\mathrm{Spec}}(R[x_1, \ldots , x_ m, y_ i, t_1, \ldots , \hat t_ i, \ldots , t_ n]) \end{align*}

which are isomorphic to $\mathbf{A}^{n + m}_ S$. In this chart the exceptional divisor is cut out by setting $y_ i = 0$ hence the exceptional divisor is smooth over $S$ as well.

Lemma 37.17.3. Let $S$ be a scheme. Let $Z \to X$ be a closed immersion of schemes smooth over $S$. Let $b : X' \to X$ be the blowing up of $Z$ with exceptional divisor $E \subset X'$. Then $X'$ and $E$ are smooth over $S$. The morphism $p : E \to Z$ is canonically isomorphic to the projective space bundle

$\mathbf{P}(\mathcal{I}/\mathcal{I}^2) \longrightarrow Z$

where $\mathcal{I} \subset \mathcal{O}_ X$ is the ideal sheaf of $Z$. The relative $\mathcal{O}_ E(1)$ coming from the projective space bundle structure is isomorphic to the restriction of $\mathcal{O}_{X'}(-E)$ to $E$.

Proof. By Divisors, Lemma 31.22.11 the immersion $Z \to X$ is a regular immmersion, hence the ideal sheaf $\mathcal{I}$ is of finite type, hence $b$ is a projective morphism with relatively ample invertible sheaf $\mathcal{O}_{X'}(1) = \mathcal{O}_{X'}(-E)$, see Divisors, Lemmas 31.32.4 and 31.32.13. The canonical map $\mathcal{I} \to b_*\mathcal{O}_{X'}(1)$ gives a closed immersion

$X' \longrightarrow \mathbf{P}\left(\bigoplus \nolimits _{n \geq 0} \text{Sym}^ n_{\mathcal{O}_ X}(\mathcal{I})\right)$

by the very construction of the blowup. The restriction of this morphism to $E$ gives a canonical map

$E \longrightarrow \mathbf{P}\left(\bigoplus \nolimits _{n \geq 0} \text{Sym}^ n_{\mathcal{O}_ Z}(\mathcal{I}/\mathcal{I}^2)\right)$

over $Z$. Since $\mathcal{I}/\mathcal{I}^2$ is finite locally free if this canonical map is an isomorphism, then the final part of the lemma holds. Having said all of this, now the question is étale local on $X$. Namely, blowing up commutes with flat base change by Divisors, Lemma 31.32.3 and we can check smoothness after precomposing with a surjective étale morphism. Thus by the étale local structure of a closed immersion of schemes over $S$ given in Lemma 37.17.1 this reduces us to the case discussed in Remark 37.17.2. $\square$

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