The Stacks project

85.12 Quasi-compact formal algebraic spaces

Here is the characterization of quasi-compact formal algebraic spaces.

Lemma 85.12.1. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. The following are equivalent

  1. the reduction of $X$ (Lemma 85.7.2) is a quasi-compact algebraic space,

  2. we can find $\{ X_ i \to X\} _{i \in I}$ as in Definition 85.7.1 with $I$ finite,

  3. there exists a morphism $Y \to X$ representable by algebraic spaces which is ├ętale and surjective and where $Y$ is an affine formal algebraic space.

Proof. Omitted. $\square$

Definition 85.12.2. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. We say $X$ is quasi-compact if the equivalent conditions of Lemma 85.12.1 are satisfied.

Lemma 85.12.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent

  1. the induced map $f_{red} : X_{red} \to Y_{red}$ between reductions (Lemma 85.7.2) is a quasi-compact morphism of algebraic spaces,

  2. for every quasi-compact scheme $T$ and morphism $T \to Y$ the fibre product $X \times _ Y T$ is a quasi-compact formal algebraic space,

  3. for every affine scheme $T$ and morphism $T \to Y$ the fibre product $X \times _ Y T$ is a quasi-compact formal algebraic space, and

  4. there exists a covering $\{ Y_ j \to Y\} $ as in Definition 85.7.1 such that each $X \times _ Y Y_ j$ is a quasi-compact formal algebraic space.

Proof. Omitted. $\square$

Definition 85.12.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. We say $f$ is quasi-compact if the equivalent conditions of Lemma 85.12.3 are satisfied.

This agrees with the already existing notion when the morphism is representable by algebraic spaces (and in particular when it is representable).

Lemma 85.12.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$ which is representable by algebraic spaces. Then $f$ is quasi-compact in the sense of Definition 85.12.4 if and only if $f$ is quasi-compact in the sense of Bootstrap, Definition 78.4.1.

Proof. This is immediate from the definitions and Lemma 85.12.3. $\square$


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