The Stacks project

Lemma 86.15.8. Consider a commutative diagram

\[ \xymatrix{ B \ar[r] & B' \\ A \ar[r] \ar[u]^\varphi & A' \ar[u]_{\varphi '} } \]

in $\textit{WAdm}^{Noeth}$ with all arrows adic and topologically of finite type. Assume $A \to A'$ flat and $B \to B'$ faithfully flat. If $\varphi '$ is rig-flat, then $\varphi $ is rig-flat.

Proof. Given $f \in A$ the assumptions of the lemma remain true for the digram

\[ \xymatrix{ B_{\{ f\} } \ar[r] & (B')_{\{ f\} } \\ A_{\{ f\} } \ar[r] \ar[u]^\varphi & (A')_{\{ f\} } \ar[u] } \]

(To check the condition on faithful flatness: faithful flatness of $B \to B'$ is equivalent to $B \to B'$ being flat and $\mathop{\mathrm{Spec}}(B'/IB') \to \mathop{\mathrm{Spec}}(B/IB)$ being surjective for some ideal of definition $I \subset A$.) Hence it suffices to prove that $\varphi $ is naively rig-flat. However, we know that $\varphi '$ is naively rig-flat and that $\mathop{\mathrm{Spec}}(B') \to \mathop{\mathrm{Spec}}(B)$ is surjective. From this the result follows immediately. $\square$

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