The Stacks project

Lemma 54.10.2. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local domain of dimension $2$. Let $A \to R$ be a surjection onto a complete discrete valuation ring. This defines a nonsingular arc $a : T = \mathop{\mathrm{Spec}}(R) \to \mathop{\mathrm{Spec}}(A)$. Let

\[ \mathop{\mathrm{Spec}}(A) = X_0 \leftarrow X_1 \leftarrow X_2 \leftarrow X_3 \leftarrow \ldots \]

be the sequence of blowing ups constructed from $a$. If $A_\mathfrak p$ is a regular local ring where $\mathfrak p = \mathop{\mathrm{Ker}}(A \to R)$, then for some $i$ the scheme $X_ i$ is regular at $x_ i$.

Proof. Let $x_1 \in \mathfrak m$ map to a uniformizer of $R$. Observe that $\kappa (\mathfrak p) = K$ is the fraction field of $R$. Write $\mathfrak p = (x_2, \ldots , x_ r)$ with $r$ minimal. If $r = 2$, then $\mathfrak m = (x_1, x_2)$ and $A$ is regular and the lemma is true. Assume $r > 2$. After renumbering if necessary, we may assume that $x_2$ maps to a uniformizer of $A_\mathfrak p$. Then $\mathfrak p/\mathfrak p^2 + (x_2)$ is annihilated by a power of $x_1$. For $i > 2$ we can find $n_ i \geq 0$ and $a_ i \in A$ such that

\[ x_1^{n_ i} x_ i - a_ i x_2 = \sum \nolimits _{2 \leq j \leq k} a_{jk} x_ jx_ k \]

for some $a_{jk} \in A$. If $n_ i = 0$ for some $i$, then we can remove $x_ i$ from the list of generators of $\mathfrak p$ and we win by induction on $r$. If for some $i$ the element $a_ i$ is a unit, then we can remove $x_2$ from the list of generators of $\mathfrak p$ and we win in the same manner. Thus either $a_ i \in \mathfrak p$ or $a_ i = u_ i x_1^{m_1} \bmod \mathfrak p$ for some $m_1 > 0$ and unit $u_ i \in A$. Thus we have either

\[ x_1^{n_ i} x_ i = \sum \nolimits _{2 \leq j \leq k} a_{jk} x_ jx_ k \quad \text{or}\quad x_1^{n_ i} x_ i - u_ i x_1^{m_ i} x_2 = \sum \nolimits _{2 \leq j \leq k} a_{jk} x_ jx_ k \]

We will prove that after blowing up the integers $n_ i$, $m_ i$ decrease which will finish the proof.

Let us see what happens with these equations on the affine blowup algebra $A' = A[\mathfrak m/x_1]$. As $\mathfrak m = (x_1, \ldots , x_ r)$ we see that $A'$ is generated over $R$ by $y_ i = x_ i/x_1$ for $i \geq 2$. Clearly $A \to R$ extends to $A' \to R$ with kernel $(y_2, \ldots , y_ r)$. Then we see that either

\[ x_1^{n_ i - 1} y_ i = \sum \nolimits _{2 \leq j \leq k} a_{jk} y_ jy_ k \quad \text{or}\quad x_1^{n_ i - 1} y_ i - u_ i x_1^{m_1 - 1} y_2 = \sum \nolimits _{2 \leq j \leq k} a_{jk} y_ jy_ k \]

and the proof is complete. $\square$

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