The Stacks project

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:

  1. 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 $\varphi ^*\mathcal{F}$.

  2. The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique isomorphism.

  3. The quasi-coherent sheaf $\mathcal{G}$ is of finite type.

  4. 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

\[ j'_* (s')^*\mathcal{G}' = s^* i'_*\mathcal{G}' = s^*\varphi ^*\mathcal{F} = t^*\varphi ^*\mathcal{F} = t^*i'_*\mathcal{G}' = j'_*(t')^*\mathcal{G}' \]

(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. $\square$

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