F}$ defines a relative almost complete intersection, then $Ext_R^n(H_0({\bf F}),R)$ is free as $S$-module and $Ext_R^k(H_0({\bf F}),R)=0$ for $k \ge n+1$.

This follows from a cohomology-and-base change argument. We assume for simplicity $S=K[[t]]$, so that we have an exact sequence $0{\longrightarrow}R \stackrel{t\cdot}{{\longrightarrow}} R {\longrightarrow}P {\longrightarrow}0$. As ${\bf F}$ defines an almost complete intersection in $R$, we have as before only two Koszul groups $H_0:=H_0({\bf F})$ and $H_1:=H_1({\bf F})$, and by assumption we have an exact sequence $$0 {\longrightarrow}H_1 \stackrel{t\cdot}{{\longrightarrow}} H_1 {\longrightarrow}\overline{H}_1 \stackrel{0}{{\longrightarrow}} H_0 \stackrel{t\cdot}{{\longrightarrow}} H_0 {\longrightarrow}\overline{H}_0 {\longrightarrow}0$$ where $\overline{H}_k:=H_k({\bf f})$. Furthermore, we have the statements from \[aci\]. We will show that $Ext^{n+1}_R(H_0,R)=0$, as are all further higher $Ext^{k}_R(H_1,R) \cong Ext^{k+2}_R(H_0,R)$, $k \ge n$. For this, note that one obtains a long exact sequence $$\ldots {\longrightarrow}E^{n+1} \stackrel{t\cdot}{{\longrightarrow}} E^{n+1} {\longrightarrow}\overline{E}^{n+1}{\longrightarrow}E^{n+2} \stackrel{t\cdot}{{\longrightarrow}} E^{n+2} {\longrightarrow}\overline{E}^{n+2}{\longrightarrow}\ldots$$ where we put temporarily $E^k:=Ext^k_R(H_0,R)$, $\overline{E}^k:=Ext_P(\overline{H}_0,P)$. As $\overline{E}^k=0$ for $k \ge n+1$ and the modules are $S$-finite, one concludes with Nakayama that all $E^{k}=0,k \ge n+1$. As $H_1=E^{n-1}$ it follows that we have an exact sequence $$0  {\longrightarrow}E^{n-1} \stackrel{t\cdot}{{\longrightarrow}} E^{n-1} {\longrightarrow}\overline{E}^{n-1} \stackrel{0}{{\longrightarrow}} E^{n} \stackrel{t \cdot}{{\longrightarrow}} E^{n} {\longrightarrow}\overline{E}^{n}{\longrightarrow}0$$ hence we find that $E^n=Ext_P^n(H_0({\bf F}),P)$ and $S$-flat, and hence $S$-free.

As in \[koszhom\] we have $$Ext^{n-1}_R(H_0({\bf F},R))=H_1({\bf F}),\;\;\;Ext^n_R(H_1({\bf F}),R)=0$$ $$0 {\longrightarrow}Ext_R^n(H_0({\bf 