The Stacks project

Remark 67.20.9. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Then $f$ has a canonical factorization

\[ Y \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y) \longrightarrow X \]

This makes sense because $f_*\mathcal{O}_ Y$ is quasi-coherent by Lemma 67.11.2. The morphism $Y \to \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y)$ comes from the canonical $\mathcal{O}_ Y$-algebra map $f^*f_*\mathcal{O}_ Y \to \mathcal{O}_ Y$ which corresponds to a canonical morphism $Y \to Y \times _ X \underline{\mathop{\mathrm{Spec}}}_ X(f_*\mathcal{O}_ Y)$ over $Y$ (see Lemma 67.20.7) whence a factorization of $f$ as above.

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