Fibrewise exact finite flat complexes lift over a Noetherian target

[base] If $e=1$, the reduced map $F_1\mathrm{/}\mathfrak{m}{F}_1\to F_0\mathrm{/}\mathfrak{m}{F}_0$ is injective by hypothesis. Apply [F1] with source $F_1$ and target $F_0$. It makes $F_1\to F_0$ injective and its cokernel flat over $R$, proving the assertion.

[IH] Suppose $e>1$ and the assertion holds for complexes with $e-1$ arrows. The reduced top arrow $F_e\mathrm{/}\mathfrak{m}{F}_e\to F_{e-1}\mathrm{/}\mathfrak{m}{F}_{e-1}$ is injective. By [F1] its lift is injective and $C=\mathrm{coker}(F_e\to F_{e-1})$ is a finite $S$-module flat over $R$. The map $F_{e-1}\to F_{e-2}$ factors through $C$, giving a shorter complex $0\to C\to F_{e-2}\to \cdots \to F_0$.

[induction] By [F2], the reduction $C\mathrm{/}\mathfrak{m}C$ is the cokernel of the reduced top arrow. Exactness of the original reduced complex therefore makes the shorter reduced complex exact at every term except possibly its last. The induction hypothesis applies, making the shorter complex exact and its final cokernel $R$-flat. Combining this with the injection $F_e\to F_{e-1}$ from step 1.2 restores exactness of the original complex; its final cokernel is the same one.

[discharge-induction: step 2.1] The base and induction steps prove the result for all $e\ge 1$. The Axiom of Choice is inherited through [F1]; all other selections are finite.