Separable finite surface extensions preserve bounded modification cohomology

Given: A finite injective local extension $A\subset B$ of permitted normal local surface domains with separable fraction-field extension $L\mathrm{/}K$ of degree $n$, assuming modification H1 over $A$ is uniformly bounded.

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-domination-of-surface-modifications-via-relative-hilbert-scheme. Assume AC and DC. Let $A$ be a normal Noetherian local domain of dimension two essentially of finite type over a field or a complete equicharacteristic Noetherian local ring. Let $B$ be a finite normal local $A$-domain, and let $Y\to \mathrm{Spec}B$ be a normal integral modification. (Finite domination of surface modifications by a relative Hilbert scheme)

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-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-trace-pairing-for-a-finite-separable-extension. Let $L\mathrm{/}F$ be a finite separable field extension. Then the bilinear pairing $L\times L\to F$, $(x,y)\mapsto {\mathrm{Tr}}_{L\mathrm{/}F}(xy)$, is nondegenerate. (The trace pairing in a finite separable extension is nondegenerate)

thm-long-exact-sequence-sheaf-cohomology. Assume the Axiom of Choice. Let $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence of abelian sheaves on a topological space $X$, and let $H^q(X,-)$ be sheaf cohomology computed from the supplied functorial injective resolution datum $I$ on $\mathrm{A}\mathrm{b}(X)$ (def-sheaf-cohomology-derived-global-sections). (Long exact sequence of sheaf cohomology)