The Stacks project

Lemma 86.15.9. The property $P(\varphi )=$“$\varphi $ is rig-flat” on arrows of $\textit{WAdm}^{Noeth}$ is a local property as defined in Formal Spaces, Remark 85.17.4.

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 85.17.1. We have to show

  1. $\varphi $ is rig-flat $\Rightarrow \varphi '$ is rig-flat,

  2. if $B \to B'$ faithfully flat, then $\varphi '$ is rig-flat $\Rightarrow \varphi $ is rig-flat, and

  3. 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 86.11.1. 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 86.15.7 and 86.15.8. We omit the trivial proof of (3). $\square$

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