The Stacks project

Proof. The pushforward $\pi _*\mathcal{G}$ of a finite type quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{G}$ is a finite type quasi-coherent $\mathcal{O}_ X$-module by Descent on Spaces, Lemma 74.6.6. In particular, if $X$ has the resolution property, then $\pi _*\mathcal{I}$ is the quotient of a finite locally free $\mathcal{O}_ X$-module by Derived Categories of Spaces, Definition 75.28.1.

Assume that we have a surjection $\mathcal{E} \to \pi _*\mathcal{I}$ for some finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$. In the rest of the proof we show that $X$ has the resolution property by reducing to the Noetherian case handled in Lemma 76.56.3. We suggest the reader skip the rest of the proof.

A first reduction is that we may view $X$ as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Spaces, Definition 65.16.2. (This doesn't affect the conditions nor the conclusion of the lemma.)

By Limits of Spaces, Lemma 70.11.3 we can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ with $Y_ i$ finite and of finite presentation over $X$ and where the transition maps are closed immersions. Consider the closed subspace $Z = V(\mathcal{I})$ of $Y$. Since $\mathcal{I}$ is of finite type, the morphism $Z \to Y$ is of finite presentation. Hence we can find an $i$ and a morphism $Z_ i \to Y_ i$ of finite presentation whose base change to $Y$ is $Z \to Y$, see Limits of Spaces, Lemma 70.7.1. For $i' \geq i$ denote $Z_{i'} = Z_ i \times _{Y_ i} Y_{i'}$. After increasing $i$ we may assume $Z_ i \to Y_ i$ is a closed immersion (of finite presentation), see Limits of Spaces, Lemma 70.6.8. Denote $\mathcal{I}_ i \subset \mathcal{O}_{Y_ i}$ the ideal sheaf of $Z_ i$ and denote $\pi _ i : Y_ i \to X$ the structure morphism. Similarly for $i' \geq i$. Since $Z = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} Z_{i'}$ we have

The transition maps in the system are all surjective as follows from the surjectivity of the maps $\pi _{i, *}\mathcal{O}_{Y_ i} \to \pi _{i', *}\mathcal{O}_{Y_{i'}}$ and the fact that $Z_{i'} = Z_ i \times _{Y_ i} Y_{i'}$. By Cohomology of Spaces, Lemma 69.5.3 for some $i' \geq i$ the map $\mathcal{E} \to \pi _*\mathcal{I}$ lifts to a map $\mathcal{E} \to \pi _{i', *}\mathcal{I}_{i'}$. After increasing $i'$ this map $\mathcal{E} \to \pi _{i', *}\mathcal{I}_{i'}$ becomes surjective (since if not the colimit of the cokernels, having surjective transition maps, is nonzero). This reduces us to the case discussed in the next paragraph.

Assume $X$ is an algebraic space over $\mathbf{Z}$ and that $Y \to X$ is of finite presentation. By absolute Noetherian approximation we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a directed limit, where each $X_ i$ is a quasi-separated algebraic space of finite type over $\mathbf{Z}$ and the transition morphisms are affine, see Limits of Spaces, Proposition 70.8.1. Since $\pi : Y \to X$ is of finite presentation we can find an $i$ and a morphism $\pi _ i : Y_ i \to X_ i$ of finite presentation whose base change to $X$ is $\pi $, see Limits of Spaces, Lemma 70.7.1. After increasing $i$ we may assume $\pi _ i$ is finite, see Limits of Spaces, Lemma 70.6.7. Next, we may assume there exists a finite locally free $\mathcal{O}_{X_ i}$-module $\mathcal{E}_ i$ whose pullback to $X$ is $\mathcal{E}$, see Limits of Spaces, Lemma 70.7.3. We may also assume there is a map $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ whose pullback to $X$ is the composition $\mathcal{E} \to \pi _*\mathcal{I} \to \pi _*\mathcal{O}_ Y$, see Limits of Spaces, Lemma 70.7.2. The cokernel

is a coherent $\mathcal{O}_{Y_ i}$-module whose pullback to $X$ is the (finitely presented) cokernel $\mathcal{Q}$ of the map $\mathcal{E} \to \pi _*\mathcal{O}_ Y$. In other words, we have $\mathcal{Q} = \pi _*(\mathcal{O}_ Y/\mathcal{I})$. Consider the map

where the second arrow is given by the algebra structure on $\pi _{i, *}\mathcal{O}_{Y_ i}$. The pullback of this map to $Y$ is zero because the image of $\mathcal{E} \to \pi _*\mathcal{O}_ Y$ is the ideal $\pi _*\mathcal{I}$. Hence by Limits of Spaces, Lemma 70.7.2 after increasing $i$ we may assume the displayed composition is zero. This exactly means that the imag of $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ is of the form $\pi _{i, *}\mathcal{I}_ i$ for some coherent ideal sheaf $\mathcal{I}_ i \subset \mathcal{O}_{Y_ i}$. Since $\mathcal{E}_ i \to \pi _{i, *}\mathcal{O}_{Y_ i}$ pulls back to $\mathcal{E} \to \pi _*\mathcal{O}_ Y$ we see that the pullback of $\mathcal{I}_ i$ to $Y$ generates $\mathcal{I}$. Denote $V_ i \subset Y_ i$ the open subspace whose complement is $V(\mathcal{I}_ i) \subset Y_ i$. Then $V$ is the inverse image of $V_ i$ by the comments above. After increasing $i$ we may assume that $V_ i$ is affine and that $\pi _ i|_{V_ i} : V_ i \to X_ i$ is étale, see Limits of Spaces, Lemmas 70.5.10 and 70.6.2. Having said all of this, we may apply Lemma 76.56.3 to conclude that $X_ i$ has the resolution property. Since $X \to X_ i$ is affine we conclude that $X$ has the resolution property too by Derived Categories of Spaces, Lemma 75.28.3. $\square$