The Stacks project

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

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). $\square$