The Stacks project

85.24 Proper morphisms

Here is the definition we will use.

Definition 85.24.1. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of formal algebraic spaces over $S$. We say $f$ is proper if $f$ is representable by algebraic spaces and is proper in the sense of Bootstrap, Definition 78.4.1.

It follows from the definitions that a proper morphism is of finite type.

Lemma 85.24.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent

  1. $f$ is proper,

  2. for every scheme $Z$ and morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ is proper,

  3. for every affine scheme $Z$ and every morphism $Z \to Y$ the base change $Z \times _ Y X \to Z$ of $f$ is proper,

  4. for every affine scheme $Z$ and every morphism $Z \to Y$ the formal algebraic space $Z \times _ Y X$ is an algebraic space proper over $Z$,

  5. there exists a covering $\{ Y_ j \to Y\} $ as in Definition 85.7.1 such that the base change $Y_ j \times _ Y X \to Y_ j$ is proper for all $j$.

Proof. Omitted. $\square$


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