Flatness over a polynomial chart from base and fibre flatness

Localize at the primes in the Statement and write $R^{\mathrm{\prime }}=R_{\mathfrak{p}}$, $P^{\mathrm{\prime }}=P_{\mathfrak{Q}}$ and $B^{\mathrm{\prime }}=B_{\mathfrak{q}}$. The hypotheses say precisely that $B^{\mathrm{\prime }}$ is $R^{\mathrm{\prime }}$-flat and that $B^{\mathrm{\prime }}\mathrm{/}\mathfrak{p}{B}^{\mathrm{\prime }}$ is flat over $P^{\mathrm{\prime }}\mathrm{/}\mathfrak{p}{P}^{\mathrm{\prime }}$. The desired conclusion is $P^{\mathrm{\prime }}$-flatness of $B^{\mathrm{\prime }}$. If $B^{\mathrm{\prime }}=0$, this is immediate; henceforth take $B^{\mathrm{\prime }}\mathrm{\ne }0$.

If $R$ is Noetherian, then $R^{\mathrm{\prime }}$, $P^{\mathrm{\prime }}$ and $B^{\mathrm{\prime }}$ are Noetherian local rings. Apply [F5] to $R^{\mathrm{\prime }}\to P^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$ with the finite $B^{\mathrm{\prime }}$-module $M=B^{\mathrm{\prime }}$. Its base-flatness and closed-fibre flatness are exactly step 1.1, so $B^{\mathrm{\prime }}$ is flat over $P^{\mathrm{\prime }}$. This proves the Noetherian case, including the situation where $B^{\mathrm{\prime }}$ is not a finite $P^{\mathrm{\prime }}$-module.

For arbitrary $R$, present the finitely presented composite $R\to B$ and the $B$-module $M=B$ as the localized directed Noetherian system of [F3]. Write its stages $R_i\to P_i\to B_i$ at the contractions of $\mathfrak{p},\mathfrak{Q},\mathfrak{q}$, with $P_i$ the localized polynomial algebra and $M_i=B_i$. The colimits are $R^{\mathrm{\prime }}\to P^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$, and each target transition is a localization of scalar extension, exactly as [F3] states. Apply [F4] to $R_i\to B_i$, since $B^{\mathrm{\prime }}$ is flat over $R^{\mathrm{\prime }}$ by step 1.1. After increasing the index, $B_i$ is flat over $R_i$; this flatness persists at every later stage by scalar extension and localization [F2].

Apply [F4] a second time to the closed-fibre system ${\bar{P}}_i=P_i\mathrm{/}{\mathfrak{p}}_i{P}_i⟶{\bar{B}}_i=B_i\mathrm{/}{\mathfrak{p}}_i{B}_i,$ where ${\mathfrak{p}}_i$ is the maximal ideal of $R_i$. These are Noetherian local maps with finite target modules. Their colimits are $P^{\mathrm{\prime }}\mathrm{/}\mathfrak{p}{P}^{\mathrm{\prime }}$ and $B^{\mathrm{\prime }}\mathrm{/}\mathfrak{p}{B}^{\mathrm{\prime }}$, because the compatible ideals ${\mathfrak{p}}_i$ have colimit $\mathfrak{p}{R}^{\mathrm{\prime }}$. For $i\le j$, the exact tensor identity is ${\bar{B}}_i{\otimes }_{{\bar{P}}_i}{\bar{P}}_j\cong B_i{\otimes }_{P_i}(P_j\mathrm{/}{\mathfrak{p}}_j{P}_j)\cong (B_i{\otimes }_{P_i}{P}_j)\mathrm{/}{\mathfrak{p}}_j(B_i{\otimes }_{P_i}{P}_j)\mathrm{.}$ Localizing at the selected target prime gives ${\bar{B}}_j$, since $B_i{\otimes }_{P_i}{P}_j\to B_j$ is a localization. The larger ideal ${\mathfrak{p}}_j$ is already killed in ${\bar{P}}_j$; using only $P_j\mathrm{/}{\mathfrak{p}}_i{P}_j$ would be a different base change. Thus the transition has exactly the localization form required by [F4]. The colimit fibre module is flat over the colimit fibre base by step 1.1. Therefore [F4] makes ${\bar{B}}_j$ flat over ${\bar{P}}_j$ at some stage $j$. Increase $j$ again if necessary to retain the base flatness from step 2.2.

At this common Noetherian stage, $B_j$ is flat over $R_j$ and $B_j\mathrm{/}{\mathfrak{p}}_j{B}_j$ is flat over $P_j\mathrm{/}{\mathfrak{p}}_j{P}_j$. Step 2.1, using [F5], makes $B_j$ flat over $P_j$. Scalar extension and localization [F2] carry this flatness to the colimit $B^{\mathrm{\prime }}$ over $P^{\mathrm{\prime }}$. Step 1.1 identifies this with the required flatness of $B_{\mathfrak{q}}$ over $P_{\mathfrak{Q}}$. The zero local algebra case was treated in step 1.1; AC is declared in [F1] and inherited through [F3]–[F5].