The Stacks project

Lemma 54.8.6. Let $A \to B$ be a finite injective local ring map of local normal Nagata domains of dimension $2$. Assume that the induced extension of fraction fields is separable. If reduction to rational singularities is possible for $A$ then it is possible for $B$.

Proof. Let $n$ be the degree of the fraction field extension $L/K$. Let $\text{Trace}_{L/K} : L \to K$ be the trace. Since the extension is finite separable the trace pairing $(h, g) \mapsto \text{Trace}_{L/K}(fg)$ is a nondegenerate bilinear form on $L$ over $K$. See Fields, Lemma 9.20.7. Pick $b_1, \ldots , b_ n \in B$ which form a basis of $L$ over $K$. By the above $d = \det (\text{Trace}_{L/K}(b_ ib_ j)) \in A$ is nonzero.

Let $Y \to \mathop{\mathrm{Spec}}(B)$ be a modification with $Y$ normal. We can find a $U$-admissible blowup $X'$ of $\mathop{\mathrm{Spec}}(A)$ such that the strict transform $Y'$ of $Y$ is finite over $X'$, see More on Flatness, Lemma 38.31.2. Picture

\[ \xymatrix{ Y' \ar[d] \ar[r] & Y \ar[r] & \mathop{\mathrm{Spec}}(B) \ar[d] \\ X' \ar[rr] & & \mathop{\mathrm{Spec}}(A) } \]

After replacing $X'$ and $Y'$ by their normalizations we may assume that $X'$ and $Y'$ are normal modifications of $\mathop{\mathrm{Spec}}(A)$ and $\mathop{\mathrm{Spec}}(B)$. In this way we reduce to the case where there exists a commutative diagram

\[ \xymatrix{ Y \ar[d]_\pi \ar[r]_-g & \mathop{\mathrm{Spec}}(B) \ar[d] \\ X \ar[r]^-f & \mathop{\mathrm{Spec}}(A) } \]

with $X$ and $Y$ normal modifications of $\mathop{\mathrm{Spec}}(A)$ and $\mathop{\mathrm{Spec}}(B)$ and $\pi $ finite.

The trace map on $L$ over $K$ extends to a map of $\mathcal{O}_ X$-modules $\text{Trace} : \pi _*\mathcal{O}_ Y \to \mathcal{O}_ X$. Consider the map

\[ \Phi : \pi _*\mathcal{O}_ Y \longrightarrow \mathcal{O}_ X^{\oplus n},\quad s \longmapsto (\text{Trace}(b_1s), \ldots , \text{Trace}(b_ ns)) \]

This map is injective (because it is injective in the generic point) and there is a map

\[ \mathcal{O}_ X^{\oplus n} \longrightarrow \pi _*\mathcal{O}_ Y,\quad (s_1, \ldots , s_ n) \longmapsto \sum b_ i s_ i \]

whose composition with $\Phi $ has matrix $\text{Trace}(b_ ib_ j)$. Hence the cokernel of $\Phi $ is annihilated by $d$. Thus we see that we have an exact sequence

\[ H^0(X, \mathop{\mathrm{Coker}}(\Phi )) \to H^1(Y, \mathcal{O}_ Y) \to H^1(X, \mathcal{O}_ X)^{\oplus n} \]

Since the right hand side is bounded by assumption, it suffices to show that the $d$-torsion in $H^1(Y, \mathcal{O}_ Y)$ is bounded. This is the content of Lemma 54.8.2 and ( $\square$

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