The Stacks project

Lemma 70.11.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume:

  1. $f$ is of locally of finite type.

  2. $X$ is quasi-compact and quasi-separated, and

  3. $Y$ is quasi-compact and quasi-separated.

Then there exists a morphism of finite presentation $f' : X' \to Y$ and a closed immersion $X \to X'$ of algebraic spaces over $Y$.

Proof. By Proposition 70.8.1 we can write $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$ with $X_ i$ quasi-separated of finite type over $\mathbf{Z}$ and with transition morphisms $f_{ii'} : X_ i \to X_{i'}$ affine. Consider the commutative diagram

\[ \xymatrix{ X \ar[r] \ar[rd] & X_{i, Y} \ar[r] \ar[d] & X_ i \ar[d] \\ & Y \ar[r] & \mathop{\mathrm{Spec}}(\mathbf{Z}) } \]

Note that $X_ i$ is of finite presentation over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Morphisms of Spaces, Lemma 67.28.7. Hence the base change $X_{i, Y} \to Y$ is of finite presentation by Morphisms of Spaces, Lemma 67.28.3. Observe that $\mathop{\mathrm{lim}}\nolimits X_{i, Y} = X \times Y$ and that $X \to X \times Y$ is a monomorphism. By Lemma 70.5.12 we see that $X \to X_{i, Y}$ is a monomorphism for $i$ large enough. Fix such an $i$. Note that $X \to X_{i, Y}$ is locally of finite type (Morphisms of Spaces, Lemma 67.23.6) and a monomorphism, hence separated and locally quasi-finite (Morphisms of Spaces, Lemma 67.27.10). Hence $X \to X_{i, Y}$ is representable. Hence $X \to X_{i, Y}$ is quasi-affine because we can use the principle Spaces, Lemma 65.5.8 and the result for morphisms of schemes More on Morphisms, Lemma 37.43.2. Thus Lemma 70.11.5 gives a factorization $X \to X' \to X_{i, Y}$ with $X \to X'$ a closed immersion and $X' \to X_{i, Y}$ of finite presentation. Finally, $X' \to Y$ is of finite presentation as a composition of morphisms of finite presentation (Morphisms of Spaces, Lemma 67.28.2). $\square$

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