A rank-one surface module is principalized by an ideal blowup

Torsion-freeness makes the natural map $M\to M{\otimes }_A\mathrm{Frac}(A)=\mathrm{Frac}(A)$ injective, so $M$ is identified with a nonzero $A$-submodule of the fraction field; choose a finite generating family $m_1,\dots ,m_n$ of $M$ over $A$.

Write $m_i=e_i\mathrm{/}{d}_i$ with $e_i\in A$ and $0\mathrm{\ne }{d}_i\in A$, put $D=d_1\cdots d_n\mathrm{\ne }0$, and let $J:=D\cdot M\subseteq A$. Then $J$ is a nonzero ideal of $A$ and multiplication by $D$ is an isomorphism of $A$-modules $M\to J$ with inverse division by $D$, which is well defined because $D\cdot M=J$ and $M$ is torsion-free.

Let $b\text{\hspace{0.17em}⁣}:Y={\mathrm{Bl}}_J\mathrm{Spec}A\to \mathrm{Spec}A$ be the blowup of $A$ along $J$; the scheme $Y$ is integral because $A$ is a domain and $J\mathrm{\ne }0$, the morphism $b$ is projective, and the pullback ideal $J{\mathcal{O}}_Y$ is invertible on $Y$.

The inclusion $J\hookrightarrow A$ of step 2.1 pulls back to a map $b^{\ast }J\to {\mathcal{O}}_Y$ of ${\mathcal{O}}_Y$-modules whose image is the invertible ideal $J{\mathcal{O}}_Y$; the map is an isomorphism at the generic point, so its kernel $T$ has zero generic stalk, that is, $T$ is a torsion ${\mathcal{O}}_Y$-module. Since $b^{\ast }M\cong b^{\ast }J$, the module $b^{\ast }M\mathrm{/}T$ is isomorphic to the invertible sheaf $J{\mathcal{O}}_Y$, so $b^{\ast }M$ modulo torsion is invertible.

On an integral affine chart $\mathrm{Spec}B\subseteq Y$ with fraction field $L$, an element of the kernel of $b^{\ast }M\to J{\mathcal{O}}_Y$ is an element of the finite $B$-module $M{\otimes }_{A}B$ killed after multiplying by some element clearing its zero generic germ, so it is annihilated by a nonzero element of $B$; conversely an element annihilated by a nonzero scalar maps to zero in the torsion-free invertible module $J{\mathcal{O}}_Y$. Hence the kernel is exactly the torsion submodule of $b^{\ast }M$ and the quotient by it is invertible.

If $g\text{\hspace{0.17em}⁣}:Z\to Y$ is an integral model dominating $Y$, then $g^{\ast }J{\mathcal{O}}_Y=J{\mathcal{O}}_Z$ is invertible as the pullback of an invertible sheaf, the pullback of the identification $b^{\ast }M\mathrm{/}T\cong J{\mathcal{O}}_Y$ presents $g^{\ast }{b}^{\ast }M$ modulo torsion as the invertible sheaf $J{\mathcal{O}}_Z$, and the same argument on integral affine charts applies verbatim.

The Axiom of Choice and the Axiom of Dependent Choice are inherited from the blowup and resolution suppliers; no general flattening or Fitting theorem is used.