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76.46 Pseudo-coherent morphisms

This section is the analogue of More on Morphisms, Section 37.60 for morphisms of schemes. The reader is encouraged to read up on pseudo-coherent morphisms of schemes in that section first.

The property “pseudo-coherent” of morphisms of schemes is étale local on the source-and-target. To see this use More on Morphisms, Lemmas 37.60.10 and 37.60.13 and Descent, Lemma 35.32.6. By Morphisms of Spaces, Lemma 67.22.1 we may define the notion of a pseudo-coherent morphism of algebraic spaces as follows and it agrees with the already existing notion defined in More on Morphisms, Section 37.60 when the algebraic spaces in question are representable.

Definition 76.46.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. We say $f$ is pseudo-coherent if the equivalent conditions of Morphisms of Spaces, Lemma 67.22.1 hold with $\mathcal{P} =$“pseudo-coherent”.

2. Let $x \in |X|$. We say $f$ is pseudo-coherent at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is pseudo-coherent.

Beware that a base change of a pseudo-coherent morphism is not pseudo-coherent in general.

Lemma 76.46.2. A flat base change of a pseudo-coherent morphism is pseudo-coherent.

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.60.3. $\square$

Lemma 76.46.3. A composition of pseudo-coherent morphisms is pseudo-coherent.

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.60.4. $\square$

Lemma 76.46.4. A pseudo-coherent morphism is locally of finite presentation.

Proof. Immediate from the definitions. $\square$

Lemma 76.46.5. A flat morphism which is locally of finite presentation is pseudo-coherent.

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.60.6. $\square$

Lemma 76.46.6. Let $f : X \to Y$ be a morphism of algebraic spaces pseudo-coherent over a base algebraic space $B$. Then $f$ is pseudo-coherent.

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.60.7. $\square$

Lemma 76.46.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $Y$ is locally Noetherian, then $f$ is pseudo-coherent if and only if $f$ is locally of finite type.

Proof. Omitted. Hint: Use the schemes version of this lemma, see More on Morphisms, Lemma 37.60.9. $\square$