Coherent dual extension supplies global Swan controlling sections

The sheaf $F=\mathcal{H}o{m}_X(Q,{\mathcal{O}}_X)$ is coherent: on an affine chart it is a submodule of a finite free module over the Noetherian ring, from a finite presentation of $Q$. Localization of that presentation identifies $F{\mathrm{\mid }}_U$ with $E$. By [F1] choose finite sections ${\psi }_j$ of $F$ generating it; they generate $E$ on $U$. No local freeness of $F$ at $S$ is asserted. Choose algebra generators $f_l$ of $A$. Include all ${\psi }_j$ and $f_l{\psi }_j$ in the indexed family.

At a smooth closed point $x$, the $f_l-f_l(x)$ span ${\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$, and the ${\psi }_j(x)$ span $E(x)$. Values can first be prescribed with linear combinations of ${\psi }_j$; the combinations $(f_l-f_l(x)){\psi }_j$ then change arbitrary derivatives without changing the value. Thus the family spans the eight-dimensional first-jet space $E_x\mathrm{/}{\mathfrak{m}}_x^2{E}_x$.

If $W$ is smooth, choose finitely many principal opens $D(g_i)\subset W$ on each of which two of the ${\psi }_j$ form a frame. Such opens cover because these values generate, and $W$ is quasi-compact. On $D(g_i)$ use the frame $g_i{\psi }_a,g_i{\psi }_b$ and the algebra generators $f_l,1\mathrm{/}{g}_i$ of $A_{g_i}$. Its controlling family consists of $g_i{\psi }_a,g_i{\psi }_b$, their $f_l$-multiples, and ${\psi }_a,{\psi }_b$. All are restrictions of global sections on $U$. Add these finitely many sections for every chart to the family in step 1.1. This proves all assertions, including that the controlling sections genuinely come from global functionals on $Q{\mathrm{\mid }}_U$.