Lemma 54.8.10. Let $p$ be a prime number. Let $A$ be a complete regular local ring of dimension $2$ and characteristic $p$. Let $L/K$ be a degree $p$ inseparable extension of the fraction field $K$ of $A$. Let $B \subset L$ be the integral closure of $A$. Then reduction to rational singularities is possible for $B$.
Proof. We have $A = k[[x, y]]$. Write $L = K[x]/(x^ p - f)$ for some $f \in A$ and denote $g \in B$ the congruence class of $x$, i.e., the element such that $g^ p = f$. By Algebra, Lemma 10.158.2 we see that $\text{d}f$ is nonzero in $\Omega _{K/\mathbf{F}_ p}$. By More on Algebra, Lemma 15.46.5 there exists a subfield $k^ p \subset k' \subset k$ with $p^ e = [k : k'] < \infty $ such that $\text{d}f$ is nonzero in $\Omega _{K/K_0}$ where $K_0$ is the fraction field of $A_0 = k'[[x^ p, y^ p]] \subset A$. Then
\[ \Omega _{A/A_0} = A \otimes _ k \Omega _{k/k'} \oplus A \text{d}x \oplus A \text{d}y \]
is finite free of rank $e + 2$. Set $\omega _ A = \Omega ^{e + 2}_{A/A_0}$. Consider the canonical map
\[ \text{Tr} : \Omega ^{e + 2}_{B/A_0} \longrightarrow \Omega ^{e + 2}_{A/A_0} = \omega _ A \]
of Lemma 54.2.4. By duality this determines a map
\[ c : \Omega ^{e + 2}_{B/A_0} \to \omega _ B = \mathop{\mathrm{Hom}}\nolimits _ A(B, \omega _ A) \]
Claim: the cokernel of $c$ is annihilated by a nonzero element of $B$.
Since $\text{d}f$ is nonzero in $\Omega _{A/A_0}$ we can find $\eta _1, \ldots , \eta _{e + 1} \in \Omega _{A/A_0}$ such that $\theta = \eta _1 \wedge \ldots \wedge \eta _{e + 1} \wedge \text{d}f$ is nonzero in $\omega _ A = \Omega ^{e + 2}_{A/A_0}$. To prove the claim we will construct elements $\omega _ i$ of $\Omega ^{e + 2}_{B/A_0}$, $i = 0, \ldots , p - 1$ which are mapped to $\varphi _ i \in \omega _ B = \mathop{\mathrm{Hom}}\nolimits _ A(B, \omega _ A)$ with $\varphi _ i(g^ j) = \delta _{ij}\theta $ for $j = 0, \ldots , p - 1$. Since $\{ 1, g, \ldots , g^{p - 1}\} $ is a basis for $L/K$ this proves the claim. We set $\eta = \eta _1 \wedge \ldots \wedge \eta _{e + 1}$ so that $\theta = \eta \wedge \text{d}f$. Set $\omega _ i = \eta \wedge g^{p - 1 - i}\text{d}g$. Then by construction we have
\[ \varphi _ i(g^ j) = \text{Tr}(g^ j \eta \wedge g^{p - 1 - i}\text{d}g) = \text{Tr}(\eta \wedge g^{p - 1 - i + j}\text{d}g) = \delta _{ij} \theta \]
by the explicit description of the trace map in Lemma 54.2.2.
Let $Y \to \mathop{\mathrm{Spec}}(B)$ be a normal modification. Exactly as in the proof of Lemma 54.8.6 we can reduce to the case where $Y$ is finite over a modification $X$ of $\mathop{\mathrm{Spec}}(A)$. By Lemma 54.4.2 we may even assume $X \to \mathop{\mathrm{Spec}}(A)$ is the result of a sequence of blowing ups in closed points. Picture:
\[ \xymatrix{ Y \ar[d]_\pi \ar[r]_-g & \mathop{\mathrm{Spec}}(B) \ar[d] \\ X \ar[r]^-f & \mathop{\mathrm{Spec}}(A) } \]
We may apply Lemma 54.2.4 to $\pi $ and we obtain the first arrow in
\[ \pi _*(\Omega ^{e + 2}_{Y/A_0}) \xrightarrow {\text{Tr}} (\Omega ^{e + 2}_{X/A_0})^{**} \xrightarrow {\varphi _ X} \omega _ X \]
and the second arrow is from Lemma 54.8.9 (because $f$ is a sequence of blowups in closed points). By duality for the finite morphism $\pi $ this corresponds to a map
\[ c_ Y : \Omega ^{e + 2}_{Y/A_0} \longrightarrow \omega _ Y \]
extending the map $c$ above. Hence we see that the image of $\Gamma (Y, \omega _ Y) \to \omega _ B$ contains the image of $c$. By our claim we see that the cokernel is annihilated by a fixed nonzero element of $B$. We conclude by Lemma 54.8.8. $\square$
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