The Stacks project

35.14 Properties of schemes local in the syntomic topology

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

Lemma 35.14.1. The property $\mathcal{P}(S) =$“$S$ is locally Noetherian and $(S_ k)$” is local in the syntomic topology.

Proof. We will check (1), (2) and (3) of Lemma 35.12.2. As a syntomic morphism is flat of finite presentation (Morphisms, Lemmas 29.30.7 and 29.30.6) 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 Cohomology of Schemes, Definition 30.11.1. Next, we show that if $S' \to S$ is a syntomic morphism of affines and $S$ has $\mathcal{P}$, then $S'$ has $\mathcal{P}$. This is Algebra, Lemma 10.163.4 (use Morphisms, Lemma 29.30.2 and Algebra, Definition 10.136.1 and Lemma 10.135.3). Finally, we show that if $S' \to S$ is a surjective syntomic morphism of affines and $S'$ has $\mathcal{P}$, then $S$ has $\mathcal{P}$. This is Algebra, Lemma 10.164.5. Thus (1), (2) and (3) of Lemma 35.12.2 hold and we win. $\square$

Lemma 35.14.2. The property $\mathcal{P}(S) =$“$S$ is Cohen-Macaulay” is local in the syntomic topology.

Proof. This is clear from Lemma 35.14.1 above since a scheme is Cohen-Macaulay if and only if it is locally Noetherian and $(S_ k)$ for all $k \geq 0$, see Properties, Lemma 28.12.3. $\square$


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