The Stacks project

Lemma 37.18.5. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian and $f$ is normal” is local in the fppf topology on the target and local in the smooth topology on the source.

Proof. We have $\mathcal{P}(f) = \mathcal{P}_1(f) \wedge \mathcal{P}_2(f) \wedge \mathcal{P}_3(f)$ where $\mathcal{P}_1(f)=$“the fibres of $f$ are locally Noetherian”, $\mathcal{P}_2(f)=$“$f$ is flat”, and $\mathcal{P}_3(f)=$“the fibres of $f$ are geometrically normal”. We have already seen that $\mathcal{P}_1$ and $\mathcal{P}_2$ are local in the fppf topology on the source and the target, see Lemma 37.18.4, and Descent, Lemmas 35.20.15 and 35.24.1. Thus we have to deal with $\mathcal{P}_3$.

Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fpqc covering of $Y$. Denote $f_ i : X_ i \to Y_ i$ the base change of $f$ by $\varphi _ i$. Let $i \in I$ and let $y_ i \in Y_ i$ be a point. Set $y = \varphi _ i(y_ i)$. Note that

\[ X_{i, y_ i} = \mathop{\mathrm{Spec}}(\kappa (y_ i)) \times _{\mathop{\mathrm{Spec}}(\kappa (y))} X_ y. \]

Hence in this situation we have that $X_ y$ is geometrically normal if and only if $X_{i, y_ i}$ is geometrically normal, see Varieties, Lemma 33.10.4. This fact implies $\mathcal{P}_3$ is fpqc local on the target.

Let $\{ X_ i \to X\} $ be a smooth covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\} $ is a smooth covering of the fibre. Hence the locality of $\mathcal{P}_3$ for the smooth topology on the source follows from Descent, Lemma 35.15.2. Combining the above the lemma follows. $\square$


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