Dualizing modules and trace pairing for normal projective surface modifications

Given: A regular Noetherian local ring $R$ of dimension two, a finite normal local $R$-domain $A$ of dimension two with $R\hookrightarrow A$ local, and a normal integral surface $X$ projective over $R$ with a proper birational map $f\text{\hspace{0.17em}⁣}:X\to \mathrm{Spec}A$.

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-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-projective-regular-local-base-coherent-duality-by-embedding. Assume AC and DC. Let $R$ be regular Noetherian local of dimension $d$, let $X$ be projective over $R$, and fix $i:X\hookrightarrow P={\mathbb{P}}_R^N$. Put $D_X=i^{!}({\mathcal{O}}_P(-N-1)[N+d])$. Then $D_X$ is a dualizing complex on $X$, with coherent biduality. (Projective coherent duality over a regular local base)

lem-finite-regular-base-algebra-dualizing-biduality. Assume AC and DC. Let $R$ be a regular Noetherian ring of finite dimension $d$, and let $B\mathrm{\ne }0$ be a module-finite $R$-algebra. For any integer $s$, $D_B=R{\mathrm{Hom}}_R(B,R[s])$ is a dualizing complex over $B$. (Dualizing biduality for finite algebras over a regular base)

lem-finite-closed-immersion-derived-coinduction-adjunction. Assume AC. For a finite homomorphism $A\to B$ of Noetherian rings and $G\in D^{+}(A)$, the complex $f^{!}G=R{\mathrm{Hom}}_A(B,G)$ has its natural $B$-action and is right adjoint to restriction of scalars. (Derived adjunction for finite rings and closed immersions)

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-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)

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-quotient-and-lifting-regularity-across-a-regular-element. Assume the Axiom of Choice (The Axiom of Choice). Let $(R,\mathfrak{m})$ be nonzero Noetherian local. If $x\in \mathfrak{m}$ is a nonzerodivisor and $R\mathrm{/}(x)$ is regular, then $R$ is regular and $x\mathrm{\notin }{\mathfrak{m}}^2$. For every nonzerodivisor $x\in \mathfrak{m}$, $\dim (R\mathrm{/}(x))=\dim R-1$. (quotient and lifting regularity across a regular element)

thm-localisation-and-polynomial-extension-of-regular-rings. Assume the Axiom of Choice (The Axiom of Choice). Localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular. Regularity can equivalently be tested at maximal ideals. For every nonzero such ring, $\mathrm{gldim}R=\dim R$, allowing infinity. (localisation and polynomial extension of regular rings)

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)

cor-dimension-of-a-finite-polynomial-ring-over-a-field. Let $k$ be a field and let $n\ge 0$. Then $\dim k[x_1,\dots ,x_n]=n\mathrm{.}$ (A polynomial ring in n variables over a field has dimension n)