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Lemma 86.15.7. 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 $A \to A'$ and $B \to B'$ are flat. Let $I \subset A$ be an ideal of definition. If $\varphi $ is rig-flat and $A/I \to A'/IA'$ is ├ętale, then $\varphi '$ is rig-flat.

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

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

Hence it suffices to prove that $\varphi '$ is naively rig-flat.

Take a rig-closed prime ideal $\mathfrak q' \subset B'$. We have to show that $(B')_{\mathfrak q'}$ is flat over $A'$. We can choose an $f \in A$ which maps to a unit of $B'/\mathfrak q'$ such that the induced prime ideal $\mathfrak p''$ of $A_{\{ f\} }$ is rig-closed, see Lemma 86.14.9. To be precise, here $\mathfrak q'' = \mathfrak q' B'_{\{ f\} }$ and $\mathfrak p'' = A_{\{ f\} } \cap \mathfrak q''$. Consider the diagram

\[ \xymatrix{ B'_{\mathfrak q'} \ar[r] & (B'_{\{ f\} })_{\mathfrak q''} \\ A \ar[r] \ar[u] & A_{\{ f\} } \ar[u] } \]

We want to show that the left vertical arrow is flat. The top horizontal arrow is faithfully flat as it is a local homomorphism of local rings and flat as $B'_{\{ f\} }$ is the completion of a localization of the Noetherian ring $B'_ f$. Similarly the bottom horizontal arrow is flat. Hence it suffices to prove that the right vertical arrow is flat. Finally, all the assumptions of the lemma remain true for the diagram

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

This reduces us to the case discussed in the next paragraph.

Take a rig-closed prime ideal $\mathfrak q' \subset B'$ and assume $\mathfrak p = A \cap \mathfrak q'$ is rig-closed as well. This implies also the primes $\mathfrak q = B \cap \mathfrak q'$ and $\mathfrak p' = A' \cap \mathfrak q'$ are rig-closed, see Lemma 86.14.4. We are going to apply Algebra, Lemma 10.100.2 to the diagram

\[ \xymatrix{ B_\mathfrak q \ar[r] & B'_{\mathfrak q'} \\ A_\mathfrak p \ar[u] \ar[r] & A'_{\mathfrak p'} \ar[u] } \]

with $M = B_\mathfrak q$. The only assumption that hasn't been checked yet is the fact that $\mathfrak p$ generates the maximal ideal of $A'_{\mathfrak p'}$. This follows from Lemma 86.14.11. $\square$


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