Fibrewise exactness of a finite free complex is open in a flat Cohen-Macaulay family

For $e=0$ the claim is immediate. Assume $e\ge 1$ and fix $\mathfrak{q}$ where the fibre complex is positively exact. Put $\mathfrak{p}=\mathfrak{q}\cap R$. The local map $R_{\mathfrak{p}}\to S_{\mathfrak{q}}$ is Noetherian, and every $F_{j,\mathfrak{q}}$ is finite over $S_{\mathfrak{q}}$ and flat over $R_{\mathfrak{p}}$ because $S$ is $R$-flat. By [F1] the total complex $F_{\bullet ,\mathfrak{q}}$ is positively exact. Its homology modules are finite over the Noetherian ring $S$, so [F3] gives $g\in S\setminus \mathfrak{q}$ such that $F_{\bullet }$ is positively exact on $D(g)$.

Put $r_i=n_i-n_{i+1}+\cdots +(-1{)}^{e-i}{n}_e$ and let $I_i\subseteq S$ be the ideal of $r_i$-minors of $d_i$. By [F2] at the total local ring $S_{\mathfrak{q}}$, all $r_i$ are nonnegative. At every point of $D(g)$, [F2] applied to the exact total complex makes all $(r_i+1)$-minors vanish in that local ring. Applying the associated-prime detection of [F3] to the Noetherian ring $S_g$ shows those larger minors vanish identically in $S_g$, including nilpotent coefficients.

Apply [F2] again, now to the fibre local ring $S_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{S}_{\mathfrak{q}}$, where the reduced complex is exact by hypothesis. For each $i$, either $(I_i)$ is the unit ideal there or it contains a regular sequence of length $i$. In the unit case, a selected $r_i$-minor is not in $\mathfrak{q}$, so on a principal neighbourhood of $\mathfrak{q}$ the ideal $I_i$ is unit in every local ring and every local fibre. In the other case, choose the regular sequence in the localized fibre ideal $(I_i{)}_{\mathfrak{q}}\mathrm{/}\mathfrak{p}(I_i{)}_{\mathfrak{q}}$. Clearing its finitely many denominators gives $f_{i1},\dots ,f_{ii}\in I_i\subseteq S$ whose images still form a regular sequence in that local fibre, since the denominators are units there.

For each nonunit case, apply [F4] to the tuple $(f_{i1},\dots ,f_{ii})$. It gives an ambient open neighbourhood $U_i$ of $\mathfrak{q}$ whose points inside $V(f_{i1},\dots ,f_{ii})$ have this tuple regular in their local fibres. At a point of $U_i$ outside that zero set, at least one $f_{ij}\in I_i$ is a unit locally, so $I_i$ is the unit ideal in the local fibre. Thus throughout $U_i$ the fibre local determinantal ideal has the unit-or-regular alternative required by [F2]. The unit cases from step 3.1 have their own principal neighbourhoods.

Intersect $D(g)$ with the finitely many open neighbourhoods from steps 3.1–4.1. At every point ${\mathfrak{q}}^{\mathrm{\prime }}$ of this open set, the fibre local complex has nonnegative expected ranks $r_i$; its larger minors vanish by step 2.1, and its $r_i$-minor ideals satisfy the unit-or-regular alternative by step 4.1. The sufficiency direction of [F2] makes that fibre local complex positively exact. Every initially exact point has such a neighbourhood, proving openness. AC is inherited through [F1]–[F4]; the neighbourhood intersection and denominator choices are finite.