The Stacks project

Theorem 76.12.8 (Grothendieck Existence Theorem). In Situation 76.12.1 there exists a finitely presented $\mathcal{O}_ X$-module $\mathcal{F}$, flat over $A$, with support proper over $A$, such that $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_{X_ n}$ for all $n$ compatibly with the maps $\varphi _ n$.

Proof. Apply Lemmas 76.12.2, 76.12.3, 76.12.4, 76.12.5, 76.12.6, and 76.12.7 to get an open subspace $W \subset X$ containing all points lying over $\mathop{\mathrm{Spec}}(A_ n)$ and a finitely presented $\mathcal{O}_ W$-module $\mathcal{F}$ whose support is proper over $A$ with $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ W} \mathcal{O}_{X_ n}$ for all $n \geq 1$. (This makes sense as $X_ n \subset W$.) By Lemma 76.3.6 we see that $\mathcal{F}$ is universally pure relative to $\mathop{\mathrm{Spec}}(A)$. By Theorem 76.11.7 (for explanation, see Lemma 76.11.8) there exists a universal flattening $S' \to \mathop{\mathrm{Spec}}(A)$ of $\mathcal{F}$ and moreover the morphism $S' \to \mathop{\mathrm{Spec}}(A)$ is a monomorphism of finite presentation. In particular $S'$ is a scheme (this follows from the proof of the theorem but it also follows a postoriori by Morphisms of Spaces, Proposition 66.50.2). Since the base change of $\mathcal{F}$ to $\mathop{\mathrm{Spec}}(A_ n)$ is $\mathcal{F}_ n$ we find that $\mathop{\mathrm{Spec}}(A_ n) \to \mathop{\mathrm{Spec}}(A)$ factors (uniquely) through $S'$ for each $n$. By More on Flatness, Lemma 38.28.8 we see that $S' = \mathop{\mathrm{Spec}}(A)$. This means that $\mathcal{F}$ is flat over $A$. Finally, since the scheme theoretic support $Z$ of $\mathcal{F}$ is proper over $\mathop{\mathrm{Spec}}(A)$, the morphism $Z \to X$ is closed. Hence the pushforward $(W \to X)_*\mathcal{F}$ is supported on $W$ and has all the desired properties. $\square$

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