Top differential lattices map into point-blowup canonical modules

Given: A regular local surface $A$ of characteristic $p$, a subring $A_0\subseteq A$ with ${\mathrm{\Omega }}_{A\mathrm{/}{A}_0}$ coherent and free of finite rank $r$, the chosen top form ${\omega }_A={\wedge }^r{\mathrm{\Omega }}_{A\mathrm{/}{A}_0}$, and a finite sequence of regular point blowups $X\to \mathrm{Spec}A$.

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-kahler-differentials-algebra. Let $A\stackrel{\phi }{\to }B$ be a homomorphism of commutative rings and let ${\mathrm{Der}}_A(B,-)$ be the derivation functor of def-derivation-algebra. (Universal Kähler differential module)

def-sheaf-relative-differentials. Let $f\text{\hspace{0.17em}⁣}:X\to S$ be a morphism of schemes (def-morphism-of-schemes), so that $f$ is in particular a morphism of ringed spaces and comes with a map of sheaves of rings $f^{\mathrm{♯}}\text{\hspace{0.17em}⁣}:f^{-1}{\mathcal{O}}_S\to {\mathcal{O}}_X$ (def-inverse-image-presheaf-and-sheaf); the pair $(X,f)$ is an $S$-scheme (def-scheme-over-base). (Sheaf of relative Kähler differentials)

thm-conormal-exact-sequence-algebra. Let $A\to P$ be a homomorphism of commutative rings, let $I\subseteq P$ be an ideal and let $B=P\mathrm{/}I$, with quotient map $\pi \text{\hspace{0.17em}⁣}:P\to B$. (Conormal exact sequence for an algebra quotient)

lem-regular-surface-reflexive-modules-and-codimension-one-lattices. Assume AC and DC. On a regular Noetherian surface a coherent reflexive module is locally free. For a finite module $M$ over a normal Noetherian domain, a generic vector belonging to $M^{\ast \ast }$ at every height-one localization belongs to $M^{\ast \ast }$. For a coherent generic-rank-$r$ module on a regular surface, $({\wedge }^{r}M{)}^{\ast \ast }$ is the determinant line of $M^{\ast \ast }$. (Reflexive surface modules and codimension-one lattice extension)

lem-regular-surface-point-blowup-canonical-transform. Assume AC and DC. For the point blowup $b:X^{\mathrm{\prime }}\to X$ of a regular Noetherian surface in the fixed regular-base dualizing setting, with exceptional divisor $E$, there is a canonical generic-compatible identification ${\omega }_{X^{\mathrm{\prime }}}=b^{\ast }{\omega }_X\otimes O_{X^{\mathrm{\prime }}}(E)$. (Canonical modules transform by the exceptional divisor at a regular point blowup)