Rational normal surface singularities and bounded modification cohomology

Assume AC and DC. A normal two-dimensional Noetherian local domain $A$ essentially of finite type over a field or complete equicharacteristic local base defines a rational singularity if $H^1(Y,{\mathcal{O}}_Y)=0$ for every normal integral proper modification $Y\to \mathrm{Spec}A$. Bounded modification H1 means these modules have uniformly bounded $A$-length. In both definitions it suffices to test projective modifications, or terminal schemes of finite normalized point-blowup sequences.

• Rationality is tested on all normal integral proper modifications, and the definition records that projective modifications and finite normalized point-blowup models suffice.
• Boundedness is a statement about the family of all modifications, not about a single one.
• The equivalence of the tests is supplied by the normalized-point domination lemma (Normalized point blowups dominate local normal surface modifications): a projective normalized point-blowup model dominating a given normal proper modification exists, the Leray injection of The Leray sequence for normal surface modifications embeds the cohomology of the given modification into that of the model, and the model has finite $A$-length by proper coherent finiteness and the codimension-one isomorphism. The restricted test classes are themselves normal proper modifications, so the reverse implication is immediate.