Scheme-theoretic image of a quasi-compact morphism

Let $V=\mathrm{Spec}A$ be an affine open of $Y$. By [F1], $f^{-1}(V)$ is quasi-compact, so its affine-open cover has a finite subcover $U_1,\dots ,U_n$; when $f^{-1}(V)$ is empty, take the finite empty cover. The finite-subcover step and the correspondence used below are licensed by the given AC.

Write $U_i=\mathrm{Spec}{B}_i$ and let ${\phi }_i:A\to B_i$ be induced by $f{\mathrm{\mid }}_{U_i}$. The sheaf condition makes $\mathrm{\Gamma }(f^{-1}(V),{\mathcal{O}}_X)$ inject into ${\prod }_i{B}_i$, so the kernel of $A\to \mathrm{\Gamma }(f^{-1}(V),{\mathcal{O}}_X)$ equals the ideal $I_V=\ker (A\to {\prod }_i{B}_i)$. For $g\in A$, the opens $U_i\cap f^{-1}(D(g))=D_{U_i}({\phi }_i(g))$ cover $f^{-1}(D(g))$, and [F4] identifies their section rings with $(B_i{)}_{{\phi }_i(g)}$. The sheaf condition therefore makes the kernel of $A_g\to \mathrm{\Gamma }(f^{-1}(D(g)),{\mathcal{O}}_X)$ equal to $\ker (A_g\to {\prod }_i(B_i{)}_{{\phi }_i(g)})$. By [F3], localizing the exact sequence $0\to I_V\to A\to {\prod }_i{B}_i$ identifies this kernel with $(I_V{)}_g$: localization commutes with the finite product. The same argument includes the empty cover, for which both kernels are the whole source ring.

Thus $\mathcal{I}{\mathrm{\mid }}_V$ is the ideal sheaf associated to $I_V$ on every affine $V$, so $\mathcal{I}$ is quasi-coherent.

By [F2], $\mathcal{I}$ defines a closed subscheme $V(\mathcal{I})$. On each affine $V$, the map $A\to {\prod }_i{B}_i$ factors through $A\mathrm{/}{I}_V$, so $f$ factors through it. If a closed subscheme defined on $V$ by $J\subseteq A$ also receives $f$, then $J\subseteq I_V$; hence $V(I_V)$ factors through that competing closed subscheme. This proves minimality.

Repeating the kernel computation on an open $W\subseteq Y$ gives the restriction of $\mathcal{I}$ to $W$. The correspondence in [F2] therefore identifies $V(\mathcal{I})\cap W$ with the scheme-theoretic image of $f^{-1}(W)\to W$.