Quotient module sheaf and its support

The identification on distinguished opens: for $f\in A$ the localisation of the exact sequence of $A$-modules $0\to I\to A\to A\mathrm{/}I\to 0$ at $f$ is the exact sequence $0\to I_f\to A_f\to (A\mathrm{/}I{)}_f\to 0$, so the induced map $(A\mathrm{/}I{)}_f\to A_f\mathrm{/}{I}_f$ is an isomorphism by [F2]; by [F1] the sections of $\widetilde{A\mathrm{/}I}$, of ${\mathcal{O}}_X$ and of $\tilde{I}$ on $D(f)$ are $(A\mathrm{/}I{)}_f$, $A_f$ and $I_f$, so the two sheaves $\widetilde{A\mathrm{/}I}$ and ${\mathcal{O}}_X\mathrm{/}\tilde{I}$ have canonically isomorphic sections on every distinguished open, compatibly with restrictions; as morphisms of ${\mathcal{O}}_X$-modules are determined by their components on the distinguished-open basis and both sides are sheaves, these identifications assemble into a canonical isomorphism $\widetilde{A\mathrm{/}I}\cong {\mathcal{O}}_X\mathrm{/}\tilde{I}$.

The support: by [F3] the stalk of ${\mathcal{O}}_X\mathrm{/}\tilde{I}$ at a prime $\mathfrak{p}$ is $A_{\mathfrak{p}}\mathrm{/}{I}_{\mathfrak{p}}$, and by step 1.1 the same is the stalk of $\widetilde{A\mathrm{/}I}$; now $A_{\mathfrak{p}}\mathrm{/}{I}_{\mathfrak{p}}=0$ exactly when $I⊈\mathfrak{p}$, because $s\in I$ with $s\mathrm{\notin }\mathfrak{p}$ makes $s\mathrm{/}1$ a unit of $A_{\mathfrak{p}}$ lying in $I_{\mathfrak{p}}$, so that $I_{\mathfrak{p}}=A_{\mathfrak{p}}$, while for $I\subseteq \mathfrak{p}$ one has $I_{\mathfrak{p}}\subseteq \mathfrak{p}{A}_{\mathfrak{p}}\mathrm{\ne }{A}_{\mathfrak{p}}$; hence the stalk at $\mathfrak{p}$ is nonzero if and only if $\mathfrak{p}\in V(I)$, that is, $\mathrm{Supp}(\widetilde{A\mathrm{/}I})=V(I)$ by [F4] and [F5]. This also agrees with $\mathrm{Supp}(\widetilde{A\mathrm{/}I})=V({\mathrm{Ann}}_A(A\mathrm{/}I))=V(I)$ from the support theorem for the finitely generated module $A\mathrm{/}I$.

The extreme ideals and the choice accounting: if $I=A$ then $A\mathrm{/}I=0$, so $\widetilde{A\mathrm{/}I}$ is the zero sheaf and ${\mathcal{O}}_X\mathrm{/}\tilde{I}={\mathcal{O}}_X\mathrm{/}{\mathcal{O}}_X=0$, while $V(A)=\mathrm{\varnothing }$ is indeed the support of the zero sheaf; if $I=0$ then $A\mathrm{/}I=A$, so $\widetilde{A\mathrm{/}I}={\mathcal{O}}_X$, while $\tilde{I}=0$ as the associated sheaf of the zero ideal, so ${\mathcal{O}}_X\mathrm{/}\tilde{I}={\mathcal{O}}_X$, and $V(0)=X$. The only appeal to the Axiom of Choice is the inherited one in [F1] and [F4], and the module $A\mathrm{/}I$ is cyclic, generated by the class of $1$, so the finite-generation hypothesis of the support theorem is met without any selection.