In this section we collect some elementary results on supports of quasi-coherent modules on algebraic spaces. Let $X$ be an algebraic space. The support of an abelian sheaf on $X_{\acute{e}tale}$ has been defined in Properties of Spaces, Section 66.20. We use the same definition for supports of modules. The following lemma tells us this agrees with the notion as defined for quasi-coherent modules on schemes.
Lemma 67.15.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $U$ be a scheme and let $\varphi : U \to X$ be an étale morphism. Then where the left hand side is the support of $\varphi ^*\mathcal{F}$ as a quasi-coherent module on the scheme $U$.
\[ \text{Supp}(\varphi ^*\mathcal{F}) = |\varphi |^{-1}(\text{Supp}(\mathcal{F})) \]
Proof. Let $u\in U$ be a (usual) point and let $\overline{x}$ be a geometric point lying over $u$. By Properties of Spaces, Lemma 66.29.4 we have $(\varphi ^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = \mathcal{F}_{\overline{x}}$. Since $\mathcal{O}_{U, u} \to \mathcal{O}_{X, \overline{x}}$ is the strict henselization by Properties of Spaces, Lemma 66.22.1 we see that it is faithfully flat (see More on Algebra, Lemma 15.45.1). Thus we see that $(\varphi ^*\mathcal{F})_ u = 0$ if and only if $\mathcal{F}_{\overline{x}} = 0$. This proves the lemma. $\square$
For finite type quasi-coherent modules the support is closed, can be checked on fibres, and commutes with base change.
Lemma 67.15.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Then
The support of $\mathcal{F}$ is closed.
For a geometric point $\overline{x}$ lying over $x \in |X|$ we have
For any morphism of algebraic spaces $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$.
\[ x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_{\overline{x}} \not= 0 \Leftrightarrow \mathcal{F}_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}} \kappa (\overline{x}) \not= 0. \]
Proof. Choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$. By Lemma 67.15.1 the inverse image of the support of $\mathcal{F}$ is the support of $\varphi ^*\mathcal{F}$ which is closed by Morphisms, Lemma 29.5.3. Thus (1) follows from the definition of the topology on $|X|$.
The first equivalence in (2) is the definition of support. The second equivalence follows from Nakayama's lemma, see Algebra, Lemma 10.20.1.
Let $f : Y \to X$ be as in (3). Note that $f^*\mathcal{F}$ is of finite type by Properties of Spaces, Section 66.30. For the final assertion, let $\overline{y}$ be a geometric point of $Y$ mapping to the geometric point $\overline{x}$ on $X$. Recall that
see Properties of Spaces, Lemma 66.29.5. Hence $(f^*\mathcal{F})_{\overline{y}} \otimes \kappa (\overline{y})$ is nonzero if and only if $\mathcal{F}_{\overline{x}} \otimes \kappa (\overline{x})$ is nonzero. By (2) this implies $x \in \text{Supp}(\mathcal{F})$ if and only if $y \in \text{Supp}(f^*\mathcal{F})$, which is the content of assertion (3). $\square$
Our next task is to show that the scheme theoretic support of a finite type quasi-coherent module (see Morphisms, Definition 29.5.5) also makes sense for finite type quasi-coherent modules on algebraic spaces.
Lemma 67.15.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. There exists a smallest closed subspace $i : Z \to X$ such that there exists a quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ with $i_*\mathcal{G} \cong \mathcal{F}$. Moreover:
If $U$ is a scheme and $\varphi : U \to X$ is an étale morphism then $Z \times _ X U$ is the scheme theoretic support of $\varphi ^*\mathcal{F}$.
The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique isomorphism.
The quasi-coherent sheaf $\mathcal{G}$ is of finite type.
The support of $\mathcal{G}$ and of $\mathcal{F}$ is $|Z|$.
Proof. Choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$. Let $R = U \times _ X U$ with projections $s, t : R \to U$. Let $i' : Z' \to U$ be the scheme theoretic support of $\varphi ^*\mathcal{F}$ and let $\mathcal{G}'$ be the (unique up to unique isomorphism) finite type quasi-coherent $\mathcal{O}_{Z'}$-module with $i'_*\mathcal{G}' = \varphi ^*\mathcal{F}$, see Morphisms, Lemma 29.5.4. As $s^*\varphi ^*\mathcal{F} = t^*\varphi ^*\mathcal{F}$ we see that $R' = s^{-1}Z' = t^{-1}Z'$ as closed subschemes of $R$ by Morphisms, Lemma 29.25.14. Thus we may apply Properties of Spaces, Lemma 66.12.2 to find a closed subspace $i : Z \to X$ whose pullback to $U$ is $Z'$. Writing $s', t' : R' \to Z'$ the projections and $j' : R' \to R$ the given closed immersion, we see that
(the first and the last equality by Cohomology of Schemes, Lemma 30.5.2). Hence the uniqueness of Morphisms, Lemma 29.25.14 applied to $R' \to R$ gives an isomorphism $\alpha : (t')^*\mathcal{G}' \to (s')^*\mathcal{G}'$ compatible with the canonical isomorphism $t^*\varphi ^*\mathcal{F} = s^*\varphi ^*\mathcal{F}$ via $j'_*$. Clearly $\alpha $ satisfies the cocycle condition, hence we may apply Properties of Spaces, Proposition 66.32.1 to obtain a quasi-coherent module $\mathcal{G}$ on $Z$ whose restriction to $Z'$ is $\mathcal{G}'$ compatible with $\alpha $. Again using the equivalence of the proposition mentioned above (this time for $X$) we conclude that $i_*\mathcal{G} \cong \mathcal{F}$.
This proves existence. The other properties of the lemma follow by comparing with the result for schemes using Lemma 67.15.1. Detailed proofs omitted. $\square$
Definition 67.15.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. The scheme theoretic support of $\mathcal{F}$ is the closed subspace $Z \subset X$ constructed in Lemma 67.15.3.
In this situation we often think of $\mathcal{F}$ as a quasi-coherent sheaf of finite type on $Z$ (via the equivalence of categories of Lemma 67.14.1).
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