The Stacks project

Lemma 29.20.2. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is locally of finite presentation.

  2. For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is of finite presentation.

  3. There exists an open covering $S = \bigcup _{j \in J} V_ j$ and open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that each of the morphisms $U_ i \to V_ j$, $j\in J, i\in I_ j$ is locally of finite presentation.

  4. There exists an affine open covering $S = \bigcup _{j \in J} V_ j$ and affine open coverings $f^{-1}(V_ j) = \bigcup _{i \in I_ j} U_ i$ such that the ring map $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_ i)$ is of finite presentation, for all $j\in J, i\in I_ j$.

Moreover, if $f$ is locally of finite presentation then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_ U : U \to V$ is locally of finite presentation.

Proof. This follows from Lemma 29.13.3 if we show that the property “$R \to A$ is of finite presentation” is local. We check conditions (a), (b) and (c) of Definition 29.13.1. By Algebra, Lemma 10.13.2 being of finite presentation is stable under base change and hence we conclude (a) holds. By the same lemma being of finite presentation is stable under composition and trivially for any ring $R$ the ring map $R \to R_ f$ is of finite presentation. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma 10.22.3. $\square$


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