The Stacks project

Lemma 35.33.4. The property of morphisms of germs

\[ \mathcal{P}((X, x) \to (S, s)) = \mathcal{O}_{S, s} \to \mathcal{O}_{X, x}\text{ is flat} \]

is étale local on the source-and-target.

Proof. Given a diagram as in Definition 35.33.1 we obtain the following diagram of local homomorphisms of local rings

\[ \xymatrix{ \mathcal{O}_{U', u'} & \mathcal{O}_{V', v'} \ar[l] \\ \mathcal{O}_{U, u} \ar[u] & \mathcal{O}_{V, v} \ar[l] \ar[u] } \]

Note that the vertical arrows are localizations of étale ring maps, in particular they are essentially of finite presentation, flat, and unramified (see Algebra, Section 10.143). In particular the vertical maps are faithfully flat, see Algebra, Lemma 10.39.17. Now, if the upper horizontal arrow is flat, then the lower horizontal arrow is flat by an application of Algebra, Lemma 10.39.10 with $R = \mathcal{O}_{V, v}$, $S = \mathcal{O}_{U, u}$ and $M = \mathcal{O}_{U', u'}$. If the lower horizontal arrow is flat, then the ring map

\[ \mathcal{O}_{V', v'} \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} \longleftarrow \mathcal{O}_{V', v'} \]

is flat by Algebra, Lemma 10.39.7. And the ring map

\[ \mathcal{O}_{U', u'} \longleftarrow \mathcal{O}_{V', v'} \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} \]

is a localization of a map between étale ring extensions of $\mathcal{O}_{U, u}$, hence flat by Algebra, Lemma 10.143.8. $\square$

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