Rationality propagates to birational local surface rings

Given: A rational permitted normal local surface domain $A$, a normal two-dimensional local domain $A\subset B$ with the same fraction field essentially of finite type over $A$, and in the bounded part a uniform bound on modification H1 over $A$.

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-finite-over-projective-noetherian-affine-base-is-projective. Assume AC. Let $R$ be Noetherian, let $X$ be projective over $R$, and let $Y\to X$ be finite. Then $Y\to X$ admits a closed immersion into one relative projective space over $X$, and $Y$ is projective over $R$. Finite compositions of projective morphisms between such schemes are projective. (Finite schemes over projective schemes are projective over a Noetherian affine base)

lem-local-normal-surface-modification-dimension-and-projective-cohomology. Assume AC and DC. Let $(A,\mathfrak{m})$ be a normal Noetherian local domain of dimension two and $f:X\to \mathrm{Spec}A$ an integral modification. Then $X$ has dimension two, all closed points have local dimension two, $f$ is an isomorphism off the closed point, $f_{\ast }{\mathcal{O}}_X={\mathcal{O}}_{\mathrm{Spec}A}$, and its special fibre has dimension at most one. (Dimension and cohomology of local normal surface modifications)

lem-local-normalized-point-blowup-sequences-spread-at-closed-points. Assume AC and DC. Let $X$ be a normal integral surface locally of finite type over a permitted base and $x\in X$ a closed point with two-dimensional local ring $B$. Any finite normalized point-blowup sequence over $\mathrm{Spec}B$ spreads to the same finite sequence of normalized blowups at closed points of $X$, unchanged off $x$. (Local normalized point sequences spread at closed surface points)

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-surface-finite-type-normalization-finite. Assume AC and DC. Every integral finite-type algebra over a field or a complete equicharacteristic Noetherian local base has finite normalization, and so do its localizations. Integral schemes of finite type over these bases consequently have finite scheme normalization. (Surface finite type normalization finite)