Dimension and cohomology of local normal surface modifications

Given: A normal Noetherian local domain $(A,\mathfrak{m})$ of dimension two, an integral modification $f\text{\hspace{0.17em}⁣}:X\to \mathrm{Spec}A$ (with $X$ projective over $A$ in the cohomological part), and a quasi-coherent sheaf $F$ on $X$.

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-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-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)

lem-cm-local-codimension-and-regular-quotient-ext-concentration. Assume the Axiom of Choice and the Axiom of Dependent Choice, inherited from the resolution and Ext suppliers 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). Let $(R,\mathfrak{m})$ be a Noetherian Cohen--Macaulay local ring of dimension $D$. (CM local codimension and Ext concentration over a regular local ring)

lem-normal-domain-implies-s-two. Assume the Axiom of Choice (The Axiom of Choice). Every commutative Noetherian integrally closed domain satisfies $(S_2)$. (normal domain implies s two)

cor-flat-local-depth-additivity. Assume the Axiom of Choice. For a flat local homomorphism $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, $\mathrm{depth}(S)=\mathrm{depth}(R)+\mathrm{depth}(S\mathrm{/}\mathfrak{m}S)\mathrm{.}$ (Depth is additive for a flat local homomorphism)

cor-field-finite-type-over-a-field-is-a-finite-extension. Let $k\subseteq K$ be a field extension. If $K$ is finitely generated as a $k$-algebra, then $K$ is a finite field extension of $k$. (A field finitely generated as a k-algebra is a finite extension of k)

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)

thm-serre-vanishing. Assume the Axiom of Choice as inherited from the cited suppliers (The Axiom of Choice). Let $A$ be a Noetherian commutative ring with $1$, let $X$ be a scheme projective over $A$ in the finite-dimensional H-projective convention (def-projective-morphism-pre-proj): the structure morphism $X\to \mathrm{Spec}A$ factors as a closed immersion (Serre vanishing for coherent sheaves and ample twists)

lem-eventual-global-generation-coherent-twists. Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a Noetherian commutative ring (def-noetherian-ring-and-module) and let $X$ be a scheme projective over $A$ in the finite-dimensional H-projective convention (def-projective-morphism-pre-proj): the structure morphism $X\to \mathrm{Spec}A$ (def-affine-scheme-spectrum) factors over (Eventual generation of coherent projective twists)

thm-cech-computes-qc-cohomology-separated-scheme-affine-cover. Assume the Axiom of Choice, inherited from sheaf cohomology. Let $X$ be a quasi-compact separated scheme (def-separated-morphism-schemes), let $U_0,\dots ,U_r$ be a finite affine open cover of $X$ and let $\mathcal{F}$ be a quasi-coherent ${\mathcal{O}}_X$-module (def-quasi-coherent-module-scheme). (Cech cohomology computes quasi-coherent cohomology on a separated scheme)

thm-integrality-and-finite-module-equivalences. Let $A\subseteq B$ be commutative rings with $A\mathrm{\ne }0$, and let $b\in B$. The following are equivalent: $b$ is integral over $A$; $A[b]$ is finitely generated as an $A$-module; and there exists a faithful $A[b]$-module that is finitely generated over $A$, where faithful means that $rM=0$ implies $r=0$ for $r\in A[b]$. See def-integral-element-and-algebraic-integer. (Integrality and finite-module characterizations for one element)

Polynomial extension increases finite Noetherian dimension by the number of variables. (A Noetherian polynomial ring has dimension one larger)