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35.15 Properties of schemes local in the smooth topology

In this section we find some properties of schemes which are local on the base in the smooth topology.

Lemma 35.15.1. The property $\mathcal{P}(S) =$“$S$ is reduced” is local in the smooth topology.

Proof. We will use Lemma 35.12.2. First we note that “being reduced” is local in the Zariski topology. This is clear from the definition, see Schemes, Definition 26.12.1. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ is reduced, then $S'$ is reduced. This is Algebra, Lemma 10.163.7. Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ is reduced, then $S$ is reduced. This is Algebra, Lemma 10.164.2. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$

slogan

Lemma 35.15.2. The property $\mathcal{P}(S) =$“$S$ is normal” is local in the smooth topology.

Proof. We will use Lemma 35.12.2. First we show “being normal” is local in the Zariski topology. This is clear from the definition, see Properties, Definition 28.7.1. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ is normal, then $S'$ is normal. This is Algebra, Lemma 10.163.9. Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ is normal, then $S$ is normal. This is Algebra, Lemma 10.164.3. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$

Lemma 35.15.3. The property $\mathcal{P}(S) =$“$S$ is locally Noetherian and $(R_ k)$” is local in the smooth topology.

Proof. We will check (1), (2) and (3) of Lemma 35.12.2. As a smooth morphism is flat of finite presentation (Morphisms, Lemmas 29.34.9 and 29.34.8) we have already checked this for “being locally Noetherian” in the proof of Lemma 35.13.1. We will use this without further mention in the proof. First we note that $\mathcal{P}$ is local in the Zariski topology. This is clear from the definition, see Properties, Definition 28.12.1. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ has $\mathcal{P}$, then $S'$ has $\mathcal{P}$. This is Algebra, Lemmas 10.163.5 (use Morphisms, Lemma 29.34.2, Algebra, Lemmas 10.137.4 and 10.140.3). Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ has $\mathcal{P}$, then $S$ has $\mathcal{P}$. This is Algebra, Lemma 10.164.6. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$

Lemma 35.15.4. The property $\mathcal{P}(S) =$“$S$ is regular” is local in the smooth topology.

Proof. This is clear from Lemma 35.15.3 above since a locally Noetherian scheme is regular if and only if it is locally Noetherian and $(R_ k)$ for all $k \geq 0$. $\square$

Lemma 35.15.5. The property $\mathcal{P}(S) =$“$S$ is Nagata” is local in the smooth topology.

Proof. We will check (1), (2) and (3) of Lemma 35.12.2. First we note that being Nagata is local in the Zariski topology. This is Properties, Lemma 28.13.6. Next, we show that if $S' \to S$ is a smooth morphism of affines and $S$ is Nagata, then $S'$ is Nagata. This is Morphisms, Lemma 29.18.1. Finally, we show that if $S' \to S$ is a surjective smooth morphism of affines and $S'$ is Nagata, then $S$ is Nagata. This is Algebra, Lemma 10.164.7. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$


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