The Stacks project

Lemma 37.18.4. 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

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

Moreover, as $\varphi _ i$ is of finite presentation the field extension $\kappa (y) \subset \kappa (y_ i)$ 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 35.13.1. $\square$


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