Trace cokernels detect and bound normal surface H1

Given: A regular Noetherian local ring $R$ of dimension two, a finite normal local $R$-domain $A$ of dimension two with $R\hookrightarrow A$ local, in the permitted class, and a projective normal modification $X\to \mathrm{Spec}A$ with $M=H^1(X,{\mathcal{O}}_X)$.

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-length-duality-over-a-regular-local-base. Assume AC and DC. Let $(R,\mathfrak{m})$ be regular Noetherian local of dimension $d$ and let $(B,\mathfrak{n})$ be a module-finite local $R$-algebra with the map local. For finite-length $B$-modules put $T(M)={\mathrm{Ext}}_R^d(M,R)$ with its natural $B$-action. (Finite-length duality over a regular local base)

lem-normal-projective-surface-dualizing-module-over-regular-local-base. Assume AC and DC. Let $R$ be a regular Noetherian local ring of dimension two, let $A$ be a finite normal local $R$-domain of dimension two, with $R\hookrightarrow A$ local, and let $X$ be a normal integral scheme of dimension two projective over $R$, with a proper birational map $f:X\to \mathrm{Spec}A$. Put ${\omega }_A={\mathrm{Hom}}_R(A,R)$. (Dualizing modules and trace pairing for normal projective surface modifications)

lem-normal-surface-modification-uniform-principal-torsion-bound. Assume AC and DC. For a normal local surface domain $A$ in the permitted finite-type class and $0\mathrm{\ne }a\in A$, the lengths of $H^1(X,{\mathcal{O}}_X)[a]$ are uniformly bounded over all normal projective modifications $X\to \mathrm{Spec}A$. (Uniform principal torsion bound for surface modification cohomology)

lem-projective-normal-surface-grauert-riemenschneider-vanishing. Assume AC and DC. Let $R$ be regular Noetherian local of dimension two, $A$ a finite normal local $R$-domain of dimension two with $R\hookrightarrow A$ local, and $X\to \mathrm{Spec}A$ a projective normal modification, in the permitted finite-type/completion class. For its normalized dualizing module ${\omega }_X$ over ${\omega }_A={\mathrm{Hom}}_R(A,R)$, $H^1(X,{\omega }_X)=0$. (Grauert–Riemenschneider vanishing for the required normal surface modifications)