Lemma 74.6.1. Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module.

## 74.6 Descent of finiteness properties of modules

This section is the analogue for the case of algebraic spaces of Descent, Section 35.7. The goal is to show that one can check a quasi-coherent module has a certain finiteness conditions by checking on the members of a covering. We will repeatedly use the following proof scheme. Suppose that $X$ is an algebraic space, and that $\{ X_ i \to X\} $ is a fppf (resp. fpqc) covering. Let $U \to X$ be a surjective étale morphism such that $U$ is a scheme. Then there exists an fppf (resp. fpqc) covering $\{ Y_ j \to X\} $ such that

$\{ Y_ j \to X\} $ is a refinement of $\{ X_ i \to X\} $,

each $Y_ j$ is a scheme, and

each morphism $Y_ j \to X$ factors though $U$, and

$\{ Y_ j \to U\} $ is an fppf (resp. fpqc) covering of $U$.

Namely, first refine $\{ X_ i \to X\} $ by an fppf (resp. fpqc) covering such that each $X_ i$ is a scheme, see Topologies on Spaces, Lemma 73.7.4, resp. Lemma 73.9.5. Then set $Y_ i = U \times _ X X_ i$. A quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ is of finite type, of finite presentation, etc if and only if the quasi-coherent $\mathcal{O}_ U$-module $\mathcal{F}|_ U$ is of finite type, of finite presentation, etc. Hence we can use the existence of the refinement $\{ Y_ j \to X\} $ to reduce the proof of the following lemmas to the case of schemes. We will indicate this by saying that “*the result follows from the case of schemes by étale localization*”.

**Proof.**
This follows from the case of schemes, see Descent, Lemma 35.7.1, by étale localization.
$\square$

Lemma 74.6.2. Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is an $\mathcal{O}_{X_ i}$-module of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

**Proof.**
This follows from the case of schemes, see Descent, Lemma 35.7.3, by étale localization.
$\square$

Lemma 74.6.3. Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a flat $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a flat $\mathcal{O}_ X$-module.

**Proof.**
This follows from the case of schemes, see Descent, Lemma 35.7.5, by étale localization.
$\square$

Lemma 74.6.4. Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite locally free $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a finite locally free $\mathcal{O}_ X$-module.

**Proof.**
This follows from the case of schemes, see Descent, Lemma 35.7.6, by étale localization.
$\square$

The definition of a locally projective quasi-coherent sheaf can be found in Properties of Spaces, Section 66.31. It is also proved there that this notion is preserved under pullback.

Lemma 74.6.5. Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a locally projective $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a locally projective $\mathcal{O}_ X$-module.

**Proof.**
This follows from the case of schemes, see Descent, Lemma 35.7.7, by étale localization.
$\square$

We also add here two results which are related to the results above, but are of a slightly different nature.

Lemma 74.6.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $f$ is a finite morphism. Then $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite type if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_ Y$-module of finite type.

**Proof.**
As $f$ is finite it is representable. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Then $U = V \times _ Y X$ is a scheme with a surjective étale morphism towards $X$ and a finite morphism $\psi : U \to V$ (the base change of $f$). Since $\psi _*(\mathcal{F}|_ U) = f_*\mathcal{F}|_ V$ the result of the lemma follows immediately from the schemes version which is Descent, Lemma 35.7.9.
$\square$

Lemma 74.6.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $f$ is finite and of finite presentation. Then $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation if and only if $f_*\mathcal{F}$ is an $\mathcal{O}_ Y$-module of finite presentation.

**Proof.**
As $f$ is finite it is representable. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Then $U = V \times _ Y X$ is a scheme with a surjective étale morphism towards $X$ and a finite morphism $\psi : U \to V$ (the base change of $f$). Since $\psi _*(\mathcal{F}|_ U) = f_*\mathcal{F}|_ V$ the result of the lemma follows immediately from the schemes version which is Descent, Lemma 35.7.10.
$\square$

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