The Stacks project

Lemma 66.13.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For every closed immersion $i : Z \to X$ the sheaf $i_*\mathcal{O}_ Z$ is a quasi-coherent $\mathcal{O}_ X$-module, the map $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective and its kernel is a quasi-coherent sheaf of ideals. The rule $Z \mapsto \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to i_*\mathcal{O}_ Z)$ defines an inclusion reversing bijection

\[ \begin{matrix} \text{closed subspaces} \\ Z \subset X \end{matrix} \longrightarrow \begin{matrix} \text{quasi-coherent sheaves} \\ \text{of ideals }\mathcal{I} \subset \mathcal{O}_ X \end{matrix} \]

Moreover, given a closed subscheme $Z$ corresponding to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ a morphism of algebraic spaces $h : Y \to X$ factors through $Z$ if and only if the map $h^*\mathcal{I} \to h^*\mathcal{O}_ X = \mathcal{O}_ Y$ is zero.

Proof. Let $U \to X$ be a surjective ├ętale morphism whose source is a scheme. Consider the diagram

\[ \xymatrix{ U \times _ X Z \ar[r] \ar[d]_{i'} & Z \ar[d]^ i \\ U \ar[r] & X } \]

By Lemma 66.12.1 we see that $i$ is a closed immersion if and only if $i'$ is a closed immersion. By Properties of Spaces, Lemma 65.26.2 we see that $i'_*\mathcal{O}_{U \times _ X Z}$ is the restriction of $i_*\mathcal{O}_ Z$ to $U$. Hence the assertions on $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ are equivalent to the corresponding assertions on $\mathcal{O}_ U \to i'_*\mathcal{O}_{U \times _ X Z}$. And since $i'$ is a closed immersion of schemes, these results follow from Morphisms, Lemma 29.2.1.

Let us prove that given a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ the formula

\[ Z(T) = \{ h : T \to X \mid h^*\mathcal{I} \to \mathcal{O}_ T \text{ is zero}\} \]

defines a closed subspace of $X$. It is clearly a subfunctor of $X$. To show that $Z \to X$ is representable by closed immersions, let $\varphi : U \to X$ be a morphism from a scheme towards $X$. Then $Z \times _ X U$ is represented by the analogous subfunctor of $U$ corresponding to the sheaf of ideals $\mathop{\mathrm{Im}}(\varphi ^*\mathcal{I} \to \mathcal{O}_ U)$. By Properties of Spaces, Lemma 65.29.2 the $\mathcal{O}_ U$-module $\varphi ^*\mathcal{I}$ is quasi-coherent on $U$, and hence $\mathop{\mathrm{Im}}(\varphi ^*\mathcal{I} \to \mathcal{O}_ U)$ is a quasi-coherent sheaf of ideals on $U$. By Schemes, Lemma 26.4.6 we conclude that $Z \times _ X U$ is represented by the closed subscheme of $U$ associated to $\mathop{\mathrm{Im}}(\varphi ^*\mathcal{I} \to \mathcal{O}_ U)$. Thus $Z$ is a closed subspace of $X$.

In the formula for $Z$ above the inputs $T$ are schemes since algebraic spaces are sheaves on $(\mathit{Sch}/S)_{fppf}$. We omit the verification that the same formula remains true if $T$ is an algebraic space. $\square$


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