Completed local degrees of finite normal surface covers

Given: A finite dominant morphism $X\to Y$ of degree $n$ between integral normal surfaces over the permitted base, with $Y$ regular, and a closed point $x\in X$ over $y\in Y$ with $\dim {\mathcal{O}}_{Y,y}=2$.

cor-equicharacteristic-complete-local-power-series-quotient. Assume the Axiom of Choice. Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian local ring, let $k=A\mathrm{/}\mathfrak{m}$, and let $e={\dim }_k(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)\mathrm{.}$ Then there is a surjective $k$-algebra homomorphism $k\mathrm{⟦}{X}_1,\dots ,X_e\mathrm{⟧}\twoheadrightarrow A\mathrm{.}$ (A complete equicharacteristic Noetherian local ring is a power-series quotient)

cor-every-system-of-parameters-is-regular-in-a-cohen-macaulay-module. Assume the Axiom of Choice (The Axiom of Choice). Every system of parameters of a nonzero finite Cohen--Macaulay module over a Noetherian local ring is a regular sequence on that module. (Every system of parameters is regular in a Cohen--Macaulay module)

cor-height-preserved-under-going-down-integral-extensions. Assume the Axiom of Choice. Let $A\subseteq B$ be an integral extension of domains with $A$ integrally closed. If $\mathfrak{q}\in \mathrm{Spec}(B)$ lies over $\mathfrak{p}:=\mathfrak{q}\cap A$ and one of the heights $\mathrm{ht}(\mathfrak{p})$ or $\mathrm{ht}(\mathfrak{q})$ is finite, then both are finite and (Under going down and incomparability, lying-over primes have the same finite height)

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)

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)

lem-surface-finite-completion-factors. Assume AC. For a finite map $R\to S$ of Noetherian rings and $\mathfrak{p}\in \mathrm{Spec}R$, $\widehat{R_{\mathfrak{p}}}{\otimes }_{R}S\cong {\prod }_{\mathfrak{q}\cap R=\mathfrak{p}}\widehat{S_{\mathfrak{q}}}$. The finitely many factors use their maximal-adic completions. Thus formal fibres for finite extensions are factors of residue-field base changes of the original formal fibres. (Surface finite completion factors)

lem-surface-regular-fibres-preserve-normality. Assume AC and DC. A flat map of Noetherian rings with regular fibres carries normality of the base to normality of the target. Consequently a normal essentially finite-type local ring over a field or complete equicharacteristic Noetherian local base has normal maximal-adic completion, which is a domain. (Surface regular fibres preserve normality)

thm-auslander-buchsbaum-formula. Assume the Axiom of Choice (The Axiom of Choice). For a nonzero finite module $M$ of finite projective dimension over a nonzero Noetherian local ring $R$, ${\mathrm{pd}}_{R}M+{\mathrm{depth}}_{R}M=\mathrm{depth}R$. Consequently such an $M$ with $\mathrm{depth}M=\mathrm{depth}R$ is free. (auslander buchsbaum formula)

thm-completion-preserves-regular-local-rings. Assume the Axiom of Choice (The Axiom of Choice). A nonzero Noetherian local ring $R$ is regular if and only if its maximal-adic completion $\widehat{R}$ is regular. (completion preserves regular local rings)