The Stacks project

Lemma 54.10.1. Let $X$ be a locally Noetherian scheme. Let

\[ (X, p) = (X_0, p_0) \leftarrow (X_1, p_1) \leftarrow (X_2, p_2) \leftarrow (X_3, p_3) \leftarrow \ldots \]

be a sequence of blowups such that

  1. $p_ i$ is closed, maps to $p_{i - 1}$, and $\kappa (p_ i) = \kappa (p_{i - 1})$,

  2. there exists an $x_1 \in \mathfrak m_ p$ whose image in $\mathfrak m_{p_ i}$, $i > 0$ defines the exceptional divisor $E_ i \subset X_ i$.

Then the sequence is obtained from a nonsingular arc $a : T \to X$ as above.

Proof. Let us write $\mathcal{O}_ n = \mathcal{O}_{X_ n, p_ n}$ and $\mathcal{O} = \mathcal{O}_{X, p}$. Denote $\mathfrak m \subset \mathcal{O}$ and $\mathfrak m_ n \subset \mathcal{O}_ n$ the maximal ideals.

We claim that $x_1^ t \not\in \mathfrak m_ n^{t + 1}$. Namely, if this were the case, then in the local ring $\mathcal{O}_{n + 1}$ the element $x_1^ t$ would be in the ideal of $(t + 1)E_{n + 1}$. This contradicts the assumption that $x_1$ defines $E_{n + 1}$.

For every $n$ choose generators $y_{n, 1}, \ldots , y_{n, t_ n}$ for $\mathfrak m_ n$. As $\mathfrak m_ n \mathcal{O}_{n + 1} = x_1\mathcal{O}_{n + 1}$ by assumption (2), we can write $y_{n, i} = a_{n, i} x_1$ for some $a_{n, i} \in \mathcal{O}_{n + 1}$. Since the map $\mathcal{O}_ n \to \mathcal{O}_{n + 1}$ defines an isomorphism on residue fields by (1) we can choose $c_{n, i} \in \mathcal{O}_ n$ having the same residue class as $a_{n, i}$. Then we see that

\[ \mathfrak m_ n = (x_1, z_{n, 1}, \ldots , z_{n, t_ n}), \quad z_{n, i} = y_{n, i} - c_{n, i} x_1 \]

and the elements $z_{n, i}$ map to elements of $\mathfrak m_{n + 1}^2$ in $\mathcal{O}_{n + 1}$.

Let us consider

\[ J_ n = \mathop{\mathrm{Ker}}(\mathcal{O} \to \mathcal{O}_ n/\mathfrak m_ n^{n + 1}) \]

We claim that $\mathcal{O}/J_ n$ has length $n + 1$ and that $\mathcal{O}/(x_1) + J_ n$ equals the residue field. For $n = 0$ this is immediate. Assume the statement holds for $n$. Let $f \in J_ n$. Then in $\mathcal{O}_ n$ we have

\[ f = a x_1^{n + 1} + x_1^ n A_1(z_{n, i}) + x_1^{n - 1} A_2(z_{n, i}) + \ldots + A_{n + 1}(z_{n, i}) \]

for some $a \in \mathcal{O}_ n$ and some $A_ i$ homogeneous of degree $i$ with coefficients in $\mathcal{O}_ n$. Since $\mathcal{O} \to \mathcal{O}_ n$ identifies residue fields, we may choose $a \in \mathcal{O}$ (argue as in the construction of $z_{n, i}$ above). Taking the image in $\mathcal{O}_{n + 1}$ we see that $f$ and $a x_1^{n + 1}$ have the same image modulo $\mathfrak m_{n + 1}^{n + 2}$. Since $x_ n^{n + 1} \not\in \mathfrak m_{n + 1}^{n + 2}$ it follows that $J_ n/J_{n + 1}$ has length $1$ and the claim is true.

Consider $R = \mathop{\mathrm{lim}}\nolimits \mathcal{O}/J_ n$. This is a quotient of the $\mathfrak m$-adic completion of $\mathcal{O}$ hence it is a complete Noetherian local ring. On the other hand, it is not finite length and $x_1$ generates the maximal ideal. Thus $R$ is a complete discrete valuation ring. The map $\mathcal{O} \to R$ lifts to a local homomorphism $\mathcal{O}_ n \to R$ for every $n$. There are two ways to show this: (1) for every $n$ one can use a similar procedure to construct $\mathcal{O}_ n \to R_ n$ and then one can show that $\mathcal{O} \to \mathcal{O}_ n \to R_ n$ factors through an isomorphism $R \to R_ n$, or (2) one can use Divisors, Lemma 31.32.6 to show that $\mathcal{O}_ n$ is a localization of a repeated affine blowup algebra to explicitly construct a map $\mathcal{O}_ n \to R$. Having said this it is clear that our sequence of blowups comes from the nonsingular arc $a : T = \mathop{\mathrm{Spec}}(R) \to X$. $\square$

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