In the article [DM] of Deligne and Mumford the notion of a normal morphism is mentioned. This is just one in a series of types1 of morphisms that can all be defined similarly. Over time we will add these in their own sections as needed.
Definition 37.20.1. Let $f : X \to Y$ be a morphism of schemes. Assume that all the fibres $X_ y$ are locally Noetherian schemes.
Let $x \in X$, and $y = f(x)$. We say that $f$ is normal at $x$ if $f$ is flat at $x$, and the scheme $X_ y$ is geometrically normal at $x$ over $\kappa (y)$ (see Varieties, Definition 33.10.1).
We say $f$ is a normal morphism if $f$ is normal at every point of $X$.
So the condition that the morphism $X \to Y$ is normal is stronger than just requiring all the fibres to be normal locally Noetherian schemes.
Lemma 37.20.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent
$f$ is normal, and
$f$ is flat and its fibres are geometrically normal schemes.
Proof. This follows directly from the definitions. $\square$
Lemma 37.20.3. A smooth morphism is normal.
Proof. Let $f : X \to Y$ be a smooth morphism. As $f$ is locally of finite presentation, see Morphisms, Lemma 29.34.8 the fibres $X_ y$ are locally of finite type over a field, hence locally Noetherian. Moreover, $f$ is flat, see Morphisms, Lemma 29.34.9. Finally, the fibres $X_ y$ are smooth over a field (by Morphisms, Lemma 29.34.5) and hence geometrically normal by Varieties, Lemma 33.25.4. Thus $f$ is normal by Lemma 37.20.2. $\square$
We want to show that this notion is local on the source and target for the smooth topology. First we deal with the property of having locally Noetherian fibres.
Lemma 37.20.4. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian” is local in the fppf topology on the source and the target.
Proof. Let $f : X \to Y$ be a morphism of schemes. Let $\{ \varphi _ i : Y_ i \to Y\} _{i \in I}$ be an fppf 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
Moreover, as $\varphi _ i$ is of finite presentation the field extension $\kappa (y_ i)/\kappa (y)$ is finitely generated. Hence in this situation we have that $X_ y$ is locally Noetherian if and only if $X_{i, y_ i}$ is locally Noetherian, see Varieties, Lemma 33.11.1. This fact implies locality on the target.
Let $\{ X_ i \to X\} $ be an fppf covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\} $ is an fppf covering of the fibre. Hence the locality on the source follows from Descent, Lemma 35.16.1. $\square$
Lemma 37.20.5. 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 37.20.4, and Descent, Lemmas 35.23.15 and 35.27.1. 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
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 35.18.2. Combining the above the lemma follows. $\square$
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