Lemma 88.15.9. The property $P(\varphi )=$“$\varphi $ is rig-flat” on arrows of $\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark 87.21.4.
Proof. Let us recall what the statement signifies. First, $\textit{WAdm}^{Noeth}$ is the category whose objects are adic Noetherian topological rings and whose morphisms are continuous ring homomorphisms. Consider a commutative diagram
\[ \xymatrix{ B \ar[r] & (B')^\wedge \\ A \ar[r] \ar[u]^\varphi & (A')^\wedge \ar[u]_{\varphi '} } \]
satisfying the following conditions: $A$ and $B$ are adic Noetherian topological rings, $A \to A'$ and $B \to B'$ are étale ring maps, $(A')^\wedge = \mathop{\mathrm{lim}}\nolimits A'/I^ nA'$ for some ideal of definition $I \subset A$, $(B')^\wedge = \mathop{\mathrm{lim}}\nolimits B'/J^ nB'$ for some ideal of definition $J \subset B$, and $\varphi : A \to B$ and $\varphi ' : (A')^\wedge \to (B')^\wedge $ are continuous. Note that $(A')^\wedge $ and $(B')^\wedge $ are adic Noetherian topological rings by Formal Spaces, Lemma 87.21.1. We have to show
$\varphi $ is rig-flat $\Rightarrow \varphi '$ is rig-flat,
if $B \to B'$ faithfully flat, then $\varphi '$ is rig-flat $\Rightarrow \varphi $ is rig-flat, and
if $A \to B_ i$ is rig-flat for $i = 1, \ldots , n$, then $A \to \prod _{i = 1, \ldots , n} B_ i$ is rig-flat.
Being adic and topologically of finite type satisfies conditions (1), (2), and (3), see Lemma 88.11.1. Thus in verifying (1), (2), and (3) for the property “rig-flat” we may already assume our ring maps are all adic and topologically of finite type. Then (1) and (2) follow from Lemmas 88.15.7 and 88.15.8. We omit the trivial proof of (3). $\square$
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