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73.6 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

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

  2. each $Y_ j$ is a scheme, and

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

  4. $\{ 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 72.7.4, resp. Lemma 72.9.5. 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”.

Lemma 73.6.1. 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.

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

Lemma 73.6.2. 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 73.6.3. 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 73.6.4. 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 65.31. It is also proved there that this notion is preserved under pullback.

Lemma 73.6.5. 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 73.6.6. 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 73.6.7. 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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