Definition 66.14.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y \subset X$ be closed subspaces corresponding to quasi-coherent ideal sheaves $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_ X$. The *scheme theoretic intersection* of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\mathcal{I} + \mathcal{J}$. Then *scheme theoretic union* of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\mathcal{I} \cap \mathcal{J}$.

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