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
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
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
This map is injective (because it is injective in the generic point) and there is a map
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
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 (54.8.2.1). \square
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