Finite free map kernels over a coherent structure sheaf are locally finite type

Induct on the target rank $q$. If $q=0$, then $\ker \mathrm{\Phi }={\mathcal{A}}^{\oplus p}$, generated by its $p$ standard sections (and equal to zero if $p=0$). If $q=1$, the images of the $p$ standard sections are a finite tuple in $\mathcal{A}$, and [F1] says its relation sheaf, which is $\ker \mathrm{\Phi }$ by [F2], is locally of finite type.

Assume $q\ge 1$ and the result for target rank $q$. Let $\mathrm{\Phi }:{\mathcal{A}}^{\oplus p}\to {\mathcal{A}}^{\oplus (q+1)}$, and let $\pi :{\mathcal{A}}^{\oplus (q+1)}\to {\mathcal{A}}^{\oplus q}$ drop the first coordinate. By the induction hypothesis, around any fixed $x\in X$ the sheaf $\ker (\pi \mathrm{\Phi })$ has finitely many generating sections $s_1,\dots ,s_r$ whose germs generate every stalk on one neighborhood $V$ of $x$.

Since $\mathrm{\Phi }(s_i)$ lies in $\ker \pi =\mathcal{A}\oplus 0^{\oplus q}$, write $\mathrm{\Phi }(s_i)=(h_i,0,\dots ,0)$ with $h_i\in \mathcal{A}(V)$. By [F1], after shrinking $V$ the relation sheaf of $(h_1,\dots ,h_r)$ is generated on $V$ by finitely many sections $t^{(1)},\dots ,t^{(m)}\in \mathcal{A}(V{)}^{\oplus r}$. For each $\mathrm{\ell }$, define ${\sigma }_{\mathrm{\ell }}={\sum }_{i=1}^r{t}_i^{(\mathrm{\ell })}{s}_i$. Then $\mathrm{\Phi }({\sigma }_{\mathrm{\ell }})=0$ because ${\sum }_i{t}_i^{(\mathrm{\ell })}{h}_i=0$.

For any $y\in V$ and any germ $v\in (\ker \mathrm{\Phi }{)}_y$, step 2.1 gives $v={\sum }_i{c}_i(s_i{)}_y$ for some germs $c_i\in {\mathcal{A}}_y$. The first coordinate of ${\mathrm{\Phi }}_y(v)=0$ says ${\sum }_i{c}_i(h_i{)}_y=0$, so $(c_i{)}_i$ lies in the relation stalk generated by the $t_y^{(\mathrm{\ell })}$. Hence $v$ is an ${\mathcal{A}}_y$-linear combination of $({\sigma }_{\mathrm{\ell }}{)}_y$. Thus these finitely many sections generate every kernel stalk on $V$; [F2] and [F3] identify the resulting subsheaf with $\ker \mathrm{\Phi }$, proving local finite type.