The Stacks project

Lemma 69.13.1. Let $S$ be a scheme. Let $f : U \to X$ be a morphism of algebraic spaces over $S$. Assume $U$ is an affine scheme, $f$ is locally of finite type, and $X$ quasi-separated and locally separated. Then there exists an immersion $U \to \mathbf{A}^ n_ X$ over $X$.

Proof. Say $U = \mathop{\mathrm{Spec}}(A)$. Write $A = \mathop{\mathrm{colim}}\nolimits A_ i$ as a filtered colimit of finite type $\mathbf{Z}$-subalgebras. For each $i$ the morphism $U \to U_ i = \mathop{\mathrm{Spec}}(A_ i)$ induces a morphism

\[ U \longrightarrow X \times U_ i \]

over $X$. In the limit the morphism $U \to X \times U$ is an immersion as $X$ is locally separated, see Morphisms of Spaces, Lemma 66.4.6. By Lemma 69.5.12 we see that $U \to X \times U_ i$ is an immersion for some $i$. Since $U_ i$ is isomorphic to a closed subscheme of $\mathbf{A}^ n_{\mathbf{Z}}$ the lemma follows. $\square$

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