Uniform principal torsion bound for surface modification cohomology

Given: A normal local surface domain $A$ in the permitted finite-type class, $0\mathrm{\ne }a\in A$, and a normal projective modification $X\to \mathrm{Spec}A$.

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-normal-surface-modification-and-normalized-point-blowup. Normal schemes. A locally Noetherian scheme is normal if every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain (def-normal-noetherian-ring). This is a local condition on the local rings and is checked on an affine open cover; it does not require the global section ring to be a domain. The empty scheme is normal vacuously. (Normal scheme modifications and normalized point blowups)

lem-normal-local-surface-radical-multiple-of-a-principal-divisor. Assume AC and DC. If $A$ is a normal two-dimensional Noetherian local domain and $0\mathrm{\ne }a\in \mathfrak{m}$, there is $c\in \mathfrak{m}$ such that $A\mathrm{/}cA$ is reduced and $a\mid c^n$ for some positive integer $n$. (Reduced Cartier multiples on a normal local 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-projective-normal-surface-modification-h1-injects-off-special-fibre. Assume AC and DC. Let $A$ be a normal two-dimensional Noetherian local domain essentially of finite type over a field or complete equicharacteristic local base, and let $X\to \mathrm{Spec}A$ be a projective normal modification. If $U$ is the inverse image of the punctured spectrum, $H^1(X,{\mathcal{O}}_X)\to H^1(U,{\mathcal{O}}_U)$ is injective. (H1 of a normal surface modification injects off its special fibre)

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)

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)

thm-proper-quasi-finite-is-finite. Assume the Axiom of Choice. Every proper quasi-finite morphism of schemes $f:X\to S$ is finite (def-proper-morphism, def-quasi-finite-morphism-schemes, def-finite-morphism-schemes). No Noetherian or nonemptiness hypothesis is imposed, and the assertion is local on the base. (A proper quasi-finite morphism is finite)