Regular local surfaces have rational modification cohomology

Given: A regular two-dimensional local domain $A$ in the permitted finite-type class.

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$. (The Axiom of Choice)

def-dependent-choice. Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when $\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$ The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

def-rational-normal-surface-singularity-and-bounded-modification-h1. 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. (Rational normal surface singularities and bounded modification cohomology)

lem-blowup-point-pushforward-vanishing. Assume the Axiom of Choice. Let $S$ be a regular surface over a field $k$ (more generally a locally Noetherian scheme of dimension two whose local rings at the center are regular of dimension two) and let $p$ be a closed point with residue field $\kappa (p)$. Let $\pi \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}\to S$ be the blowup of $p$ with exceptional curve $E$. (Pushforward and vanishing for point blowups on a surface)

lem-normal-surface-modification-leray-short-exact-sequence. Assume AC and DC. Let $A$ be a normal local domain of dimension two in the field/complete-equicharacteristic finite-type class, and $X^{\mathrm{\prime }}\stackrel{g}{\to }X\to \mathrm{Spec}A$ normal integral modifications. Then $g_{\ast }{\mathcal{O}}_{X^{\mathrm{\prime }}}={\mathcal{O}}_X$ and $H^1(X,{\mathcal{O}}_X)\to H^1(X^{\mathrm{\prime }},{\mathcal{O}}_{X^{\mathrm{\prime }}})$ is injective. (The Leray sequence for normal surface modifications)

lem-normalized-point-blowups-dominate-local-normal-surface-modifications. Assume AC and DC. Let $A$ be a normal two-dimensional Noetherian local domain essentially of finite type over a field or a complete equicharacteristic Noetherian local ring. Let $S$ be a normal integral modification of $\mathrm{Spec}A$, and let $Y\to S$ be an integral modification with normal $Y$. (Normalized point blowups dominate local normal surface modifications)