Canonical adjunction for a Cartier fibre curve

Given: A regular Noetherian local ring $R$ of dimension two, a finite normal local $R$-domain $A$, a normal projective modification $X$ over $A$, a Cartier closed fibre $E\subseteq X$ with residue field $\kappa$ and conormal $L={\mathcal{O}}_X(-E){\mathrm{\mid }}_E$, and the regular-base dualizing module ${\omega }_X$ of $X$.

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)

lem-normal-projective-surface-dualizing-module-over-regular-local-base. Assume AC and DC. Let $R$ be a regular Noetherian local ring of dimension two, let $A$ be a finite normal local $R$-domain of dimension two, with $R\hookrightarrow A$ local, and let $X$ be a normal integral scheme of dimension two projective over $R$, with a proper birational map $f:X\to \mathrm{Spec}A$. Put ${\omega }_A={\mathrm{Hom}}_R(A,R)$. (Dualizing modules and trace pairing for normal projective surface modifications)

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-finite-length-duality-over-a-regular-local-base. Assume AC and DC. Let $(R,\mathfrak{m})$ be regular Noetherian local of dimension $d$ and let $(B,\mathfrak{n})$ be a module-finite local $R$-algebra with the map local. For finite-length $B$-modules put $T(M)={\mathrm{Ext}}_R^d(M,R)$ with its natural $B$-action. (Finite-length duality over a regular local base)

thm-serre-duality-for-coherent-sheaves-on-projective-cm-scheme. Assume AC. Let $k$ be a field and let $X$ be a projective, pure $d$-dimensional Cohen–Macaulay $k$-scheme. Let $D_X={\omega }_X[d]$ be its normalized dualizing complex, and let $t_X:H^d(X,{\omega }_X)\to k$ be its trace. (Serre duality for coherent sheaves on a projective Cohen–Macaulay scheme)

lem-regular-base-dualizing-traces-compose-on-rational-modifications. Assume AC and DC. Let $R$ be regular local of dimension two, $A$ finite normal local over $R$, and let $g:X^{\mathrm{\prime }}\to X$ be a morphism of projective normal modifications over $A$. Their regular-base dualizing complexes are independent of the chosen projective embeddings up to the unique isomorphism preserving their duality pairings. (Dualizing traces compose and become isomorphisms on rational modifications)

def-serre-r-k-and-s-k-conditions. For a commutative Noetherian ring $R$ and an integer $j\ge 0$, condition $(R_j)$ means that $R_{\mathfrak{p}}$ is regular whenever $\mathrm{ht}\mathfrak{p}\le j$. Condition $(S_j)$ means that $\mathrm{depth}{R}_{\mathfrak{p}}\ge \min \{j,\dim R_{\mathfrak{p}}\}$ for every prime $\mathfrak{p}$. (serre r k and s k conditions)

cor-serre-normality-criterion-two-directions. Assume the Axiom of Choice (The Axiom of Choice). A commutative Noetherian domain is normal if and only if it satisfies $(R_1)$ and $(S_2)$. Equivalently its integral closedness is characterized by these two conditions. (serre normality criterion two directions)