The Leray sequence for normal surface modifications

Given: A normal local domain $A$ of dimension two in the field or complete-equicharacteristic finite-type class, and normal integral modifications $X^{\mathrm{\prime }}\stackrel{g}{\to }X\to \mathrm{Spec}A$.

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. (def-normal-surface-modification-and-normalized-point-blowup)

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-surface-flat-base-change-coherent-cohomology-by-cech. Assume AC and DC. For a quasi-compact separated scheme $X$ over a ring $A$, a quasi-coherent sheaf $F$ and a flat $A$-algebra $C$, the canonical maps $H^q(X,F){\otimes }_{A}C\to H^q(X_C,F_C)$ are isomorphisms for every $q$. No flatness of $F$ over $A$ is required. (Flat base change for quasi-coherent surface cohomology by Čech)

lem-surface-modification-isomorphism-in-codimension-one. Assume AC. Let $f:X\to S$ be a modification of integral Noetherian schemes and let $S$ be normal of dimension two. Then $f$ is an isomorphism over an open subset containing every point of codimension at most one in $S$. The complement is a finite set of closed points. If every fibre is zero-dimensional, $f$ is an isomorphism. (A normal-surface modification is an isomorphism in codimension one)

thm-leray-spectral-sequence-for-sheaf-cohomology. Assume the Axiom of Choice. Let $f:X\to Y$ be a morphism of schemes (more generally a continuous map of topological spaces) and let $\mathcal{F}$ be an abelian sheaf on $X$. (Leray spectral sequence for sheaf cohomology)

thm-proper-pushforward-coherent. Assume the Axiom of Choice and the Axiom of Dependent Choice, inherited from the affine localization theorem, the Čech comparison and the dévissage lemma cited below (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). (Coherent higher direct images under proper morphisms)