The Stacks project

Lemma 37.20.11. The property $\mathcal{P}(f)=$“the fibres of $f$ are locally Noetherian and $f$ is Cohen-Macaulay” is local in the fppf topology on the target and local in the syntomic topology on the source.

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

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

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

and that $\kappa (y) \subset \kappa (y_ i)$ is a finitely generated field extension. Hence if $X_ y$ is locally Noetherian, then $X_{i, y_ i}$ is locally Noetherian, see Varieties, Lemma 33.11.1. And if in addition $X_ y$ is Cohen-Macaulay, then $X_{i, y_ i}$ is Cohen-Macaulay, see Varieties, Lemma 33.13.1. Thus $\mathcal{P}_2$ is fppf local on the target.

Let $\{ X_ i \to X\} $ be a syntomic covering of $X$. Let $y \in Y$. In this case $\{ X_{i, y} \to X_ y\} $ is a syntomic covering of the fibre. Hence the locality of $\mathcal{P}_2$ for the syntomic topology on the source follows from Descent, Lemma 35.14.2. Combining the above the lemma follows. $\square$


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