Canonical modules transform by the exceptional divisor at a regular point blowup

Given: The point blowup $b\text{\hspace{0.17em}⁣}:X^{\mathrm{\prime }}\to X$ of a regular Noetherian surface $X$ at a closed point $x$ in the fixed regular-base dualizing setting, with exceptional divisor $E$, and ${\omega }_X$ the regular-base canonical module.

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-affine-point-blowup-pushforward-vanishing. Assume the Axiom of Choice, inherited from the Proj construction and from the cohomology suppliers (The Axiom of Choice). (Pushforward and vanishing for an affine point blowup)

lem-relative-projective-space-derived-duality-regular-local-base. Assume AC and DC. Let $R$ be regular Noetherian local of finite dimension and $P={\mathbb{P}}_R^N$. Write $W={\mathcal{O}}_P(-N-1)$. Laurent residue gives $t:R\mathrm{\Gamma }(P,W[N])\to R$. For every $K\in D_{\mathrm{C}\mathrm{o}\mathrm{h}}^b(P)$, evaluation followed by $t$ gives a natural quasi-isomorphism $R\mathrm{\Gamma }(P,\mathcal{R}\text{\hspace{0.17em}⁣}Ho{m}_P(K,W[N]))\cong R{\mathrm{Hom}}_R(R\mathrm{\Gamma }(P,K),R)$. (Relative derived duality on projective space over a regular local ring)

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-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-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)