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

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

$\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 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). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)